OWER UNCTIONS f xax b -...

Name: ____________________________________ Date: __________________

COMMON CORE ALGEBRA II, UNIT #10 – POLYNOMIAL AND RATIONAL FUNCTIONS – LESSON #1 eMATHINSTRUCTION, RED HOOK, NY 12571, © 2015

POWER FUNCTIONS COMMON CORE ALGEBRA II

Before we start to analyze polynomials of degree higher than two (quadratics), we first will look at very simple functions known as power functions. The formal definition of a power function is given below: Exercise #1: For each of the following power functions, state the value of a and b by writing the equation in the form by ax .

(a) 2

3y

x (b)

3

1

7y

x (c) 8y x (d)

3

6y

x

The characteristics of power functions depend on both the value of a and the value of b. The most important, though, is the exponent (the a is simply a vertical stretch of the power function).

Exercise #2: Consider the general power function by ax .

For now we will just concentrate on power function where the exponent is a positive whole number. Exercise #3: Using your table, fill in the following values for common power functions.

POWER FUNCTIONS

Any function of the form: bf x ax where a and b are real numbers not equal to zero.

(a) What can be said about the y-intercept of any power function if 0b ? Illustrate.

(b) What can be said about the y-intercept of any power function if 0b ? Illustrate.

x 3 2 1 0 1 2 3

2x

3x

4x

5x


From the previous exercise, we should note that when the power function has an even exponent, then positive and negative outputs have the same value. When the power function has an odd exponent, then positive and negative inputs have opposite outputs. Recall this is the definition of even and odd functions.

Exercise #4: Using your calculators, sketch the power functions below using the standard viewing window.

(a) 2y x (b) 3y x (c) 4y x (d) 5y x Exercise #5: Which of the following power functions is shown in the graph below? Explain your choice. Do without the use of your calculator. (1) 74y x (3) 86y x (2) 103y x (4) 95y x The End Behavior of Polynomials – The behavior of polynomials as the input variable gets very large, both positive and negative, is important to understand. We will explore this in the next exercise.

Exercise #6: Consider the two functions 3 21 2 29 30y x x x and 3

2y x .

(a) Graph these functions using min max min max10, 10, 100, 100x x y y

(b) Graph these functions using min max min max20, 20, 1000, 1000x x y y

(c) Graph these functions using min max min max50, 50, 10000, 10000x x y y

(d) Graph these functions using min max min max100, 100, 100000, 100000x x y y

(e) What do you observe about the nature of the two graphs as the viewing window gets larger? (f) Why is this occurring? The end behavior of any polynomial is dictated by its highest powered term!!!

x

y

x

y

x

y

x

y

x

y

Name: ____________________________________ Date: __________________


POWER FUNCTIONS COMMON CORE ALGEBRA II HOMEWORK

FLUENCY

1. Without using your calculator, determine which of the following equations could represent the graph shown

below. Explain your choice. (1) 2y x (2) 3y x (3) 4y x (4) 5y x 2. Identify which of the following are power functions. For each that you are power function, write them in the

form ny ax , where a and n are real numbers. Placing them in these forms may take some mindful algebraic manipulation.

(a) 53y x (b) 54 7y x (c) 5

10y

x (d)

7

3

6

2

xy

x

(e) 2 2 7y x x (f) 748y x (g) 4

25y

x (f) 2

2 3y x

3. If the point (-3, 8) lies on the graph of a power function with an even exponent, which of the following

points must also lie its graph? (1) 3, 8 (3) 3, 8

(2) 3, 8 (4) 8, 3

x

y


4. If the point (-5, 12) lies on the graph of a power function with an odd exponent, which of the following points must also lie on its graph?

(1) 5, 12 (2) 12, 5

(3) 5,12 (3) 12, 5

5. For each of the following polynomials, give a power functions that best represents the end behavior of the

polynomial.

(a) 33 2 12y x x (b) 210 8y x (c) 5 36 4 120y x x x (d) 5 43 2 4 7y x x x (e) 4 25 2y x x (f) 5 7 34 8 2 3y x x x 6 The graph below could be the long-run behavior for which of the following functions? Do this problem

without graphing each of the following equations. (1) y x x 2 7 12

(2) y x x x 4 2 6 43 2

(3) y x x x x 5 3 2 94 3 2

(4) y x x x 3 4 2 15 2 REASONING

7. Let's examine why end-behavior works a little more closely. Consider the functions 3f x x and

3 22 7 10g x x x x .

(a) Fill out the table below for the values of x listed. Round your final column to the nearest hundredth.

(b) What number is the ratio in the fourth column approaching as x gets larger? What does this tell you

about the part of g x that can be attributed to the cubic term?

x

y

x f x g x

f x

g x

5

10

50

100

Name: ___________________________________ Date: _________________


GRAPHS AND ZEROES OF A POLYNOMIAL COMMON CORE ALGEBRA II

A polynomial is a function consisting of terms that all have whole number powers. In its most general form, a polynomial can be written as:

11 1 0

n nn ny a x a x a x a

Quadratic and linear functions are the simplest of all polynomials. In this lesson we will explore cubic and

quartic functions, those whose highest powers are 3 4and x x respectively.

Exercise #1: For each of the following cubic functions, sketch the graph and circle its x-intercepts.

(a) 3 23 6 8y x x x (b) 32 8 9y x x (c) 3 22 12 18y x x x

Clearly, a cubic may have one, two or three real roots and can have two turning points. Just as with parabolas, there exists a tie between a cubic’s factors and its x-intercepts.

Exercise #2: Consider the cubic whose equation is 3 2 12y x x x .

y

x

20

5

20

5

y

x

20

5

20

5

y

x

20

5

20

5

(a) Algebraically determine the zeroes of this function.

(b) Sketch a graph of this function on the axes below illustrating your answer to part (a).

y

x

7 7

25

25


Exercise #3: The largest root of 3 29 12 22 0x x x falls between what two consecutive integers?

(1) 4 and 5 (3) 10 and 11

(2) 6 and 7 (4) 8 and 9

Exercise #4: Consider the quartic function 4 25 4y x x .

Exercise #5: Consider the quartic whose equation is 4 3 23 35 39 70y x x x x .

y

x

500

500

10 10

(a) Sketch a graph of this quartic on the axes below. Label its x-intercepts.

(b) Based on your graph from part (a), write the

expression 4 3 23 35 39 70x x x x in its factored form.

y

x

5

5

10

5

(a) Algebraically determine the x-intercepts of this function.

(b) Verify your answer to part (a) by sketching a graph of the function on the axes below.

Name: ___________________________________ Date: _________________


GRAPHS AND ZEROES OF A POLYNOMIAL COMMON CORE ALGEBRA II HOMEWORK

FLUENCY

1. Consider the cubic function 3 22 8y x x x .

2. Consider the cubic function 3 22 36 72y x x x .

(a) Find an appropriate y-window for the x-window shown on the axes and sketch the graph. Make the sure the window is sufficiently large to show the two turning points and all intercepts. Clearly label all x-intercepts.

(b) What are the solutions to the equation 3 22 36 72 0x x x ?

(c) Based on your answers to (b), how must the

expression 3 22 36 72x x x factor?

y

x

10 10

(b) Sketch the function on the axes given. Clearly plot and label each x-intercept.

(a) Algebraically determine the zeroes of this cubic function..

y

x

5 5

50

50


3. Consider the cubic function given by 3 26 12 5y x x x .

4. Consider the quartic function 4 3 227 25 50y x x x x .

5. In general, how does the number of zeroes (or x-intercepts) relate to the highest power of a polynomial? Be

specific. Can you make a statement about the minimum number of zeroes as well as the maximum?

(a) Sketch the graph of this function on the axes given below. Clearly mark all x-intercepts.

(b) Use your graph from part (a) to solve the

equation 4 3 227 25 50 0x x x x .

(c) Considering your answer to (b), how does the

expression 4 3 227 25 50x x x x factor?

y

x

7 7

200

200

(a) Sketch a graph of this function on the axes given below.

(b) Considering the graphs of cubics you saw in class and those in problems 1 and 2, what is different about the way this cubic’s graph looks compared to the others? 20

20

2 6

y

x

Name: ___________________________________ Date: _________________


CREATING POLYNOMIAL EQUATIONS COMMON CORE ALGEBRA II

The connection between the zeroes of a polynomial and its factors should now be clear. This connection can be used to create equations of polynomials. The key is utilizing the factored form of a polynomial.

Exercise #1: Determine the equation of a quadratic function whose roots are 3 and 4 and which passes

through the point 2, 50 . Express your answer in standard form ( 2y ax bx c ). Verify your answer by

creating a sketch of the function on the axes below. It’s important to understand how the a value effects the graph of the polynomial. This is easiest to explore if the polynomial remains in factored form.

Exercise #2: Consider quadratic polynomials of the form 2 5y a x x , where 0a .

(a) What are the x-intercepts of this parabola? (b) Sketch on the axes given the following equations:

2 5

2 2 5

4 2 5

y x x

y x x

y x x

THE FACTORED FORM OF A POLYNOMIAL

If the set 1 2 3, , , ..., nr r r r represent the roots (zeroes) of a polynomial, then the polynomial can be written as:

1 2 ny a x r x r x r where a is some constant determined by another point

y

x

7 7

100

100

y

x

5 9

50

50


As we can see from this exercise, the value of a does not change the zeroes of the function, but does vertically stretch the function. We can create equations of higher powered polynomials in a similar fashion. Exercise #3: Create the equation of the cubic, in standard form, that has x-intercepts of 4, 2, and 5 and passes

through the point 6, 20 . Verify your answer by sketching the cubic’s graph on the axes below.

Exercise #4: Create the equation of a cubic in standard form that has a double zero at 2 and another zero at 4. The cubic has a y-intercept of 16. Sketch your cubic on the axes below to verify your result. Exercise #5: How would you describe this cubic curve at its double root?

y

x

7 7

50

50

y

x

7 7

100

100

Name: ___________________________________ Date: _________________


CREATING EQUATIONS OF POLYNOMIALS COMMON CORE ALGEBRA II HOMEWORK

FLUENCY 1. Create the equation of a quadratic polynomial, in standard form, that has zeroes of 5 and 2 and which

passes through the point 3, 24 . Sketch the graph of the quadratic below to verify your result.

2. Create the equation of a quadratic function, in standard form, that has one zero of 3 and a turning point at

1, 16 . Hint – try to determine the second zero of the parabola by thinking about the relationship

between the first zero and the turning point (axis of symmetry). Sketch your solution below.

y

x

7 7

50

50

y

x

20

5

20

5


3. Create an equation for a cubic function, in standard form, that has x-intercepts given by the set 3,1, 7

and which passes through the point 2, 54 . Sketch your result on the axes shown below.

4. Create the equation of a cubic whose x-intercepts are given by the set 6, 3, 5 and which passes through

the point 3, 36 . Note that your leading coefficient in this case will be a non-integer. Sketch your result

below.

x

5 9

y 200

200

y

x

7 7

50

20

Name: ____________________________________ Date: __________________


POLYNOMIAL IDENTITIES COMMON CORE ALGEBRA II

Polynomials are expressions consisting of addition and subtraction of variables and coefficients all raised to non-negative, integer powers. As in the last few lessons, there can be a single variable or multiple variables. Exercise #1: Which of the following is not a polynomial expression? Explain why your choice fails to be a polynomial.

(1) 3 2 32x x y y (3) 14 2 22x y xy

(2) 7 7x y (4) 2 2x y Because polynomials consist of basic operations on variables, they can be manipulated using the associative, commutative, and distributive properties (as you have done many times). These operations can result in what are known as polynomial identities. An identity is defined more broadly below:

Exercise #2: One identity that you should be familiar with is 2 2x y x y x y .

Exercise #3: Prove the identity 2 2 22a b a ab b and use it to evaluate 235 .

IDENTITIES

An identity is an equation that is true for all values of the replacement variable or variables.

(a) Test this identity with the pair 10x and 3y .

(b) Prove this identity by manipulating the right side of the equation.

(c) Use this identity to evaluate the difference 2 250 49 .

(d) Use this identify to simplify and then evaluate the product 51 49 .


Sometimes identities can have geometric connections as well as algebraic. The Pythagorean Theorem gives us ample identities. Exercise #4: A right triangle is shown below whose sides are 2 22 , 1, and 1x x x . (a) Show that these will be the side lengths of a right triangle as

long as 1x , i.e. show that

2 22 2 22 1 1x x x

is an identity.

(b) Based on our work from (a) and on the triangle shown, explain why any even integer must be part of a

Pythagorean triple, i.e. a set of 3 integers that could be the sides of a right triangle. Generate a Pythagorean triple that has 10 in it and a separate one that has 14 in it.

Exercise #5: Consider the polynomial identity 3 3 2 2 33 3a b a a b ab b .

2x

2 1x

2 1x

(a) Prove this identity by expanding the left-hand side of the equation.

(b) Use your calculator to find the value of 311 then use the identity to show the same result. Carefully consider your choice of a and b.

Name: ____________________________________ Date: __________________


POLYNOMIAL IDENTITIES COMMON CORE ALGEBRA II HOMEWORK

FLUENCY

1. One of the two expressions below is an identity and one of them is not. Determine which is an identity by

testing the truth value of the equation for various values of x. Show the values of x that you test. Remember, and identity will be true for every value of x.

Equation #1: 21 4 4x x Equation #2: 2 22 4 4x x x

2. Which of the following equations represents an identity? (1) 2 1 3 4x x (3) 4 3 2 2 7x x

(2) 6 3 2 10x x (4) 4 5 2 20 8x x

3. One of the more useful identities that students almost inherently learn is:

2x c x d x c d x cd

(a) Prove this identity. You may choose to algebraically manipulate one or both sides of the equation to justify the equivalence.

(b) This identity allows you to multiply common binomials very quickly. Find the following products in

simplest trinomial form. (i) 3 7x x (ii) 7 2x x (iii) 10 3x x


4. You should be well aware of the difference of perfect squares, i.e. 2 2x y x y x y . But there is also

an identity for the difference of perfect cubes:

3 3 2 2x y x y x xy y

APPLICATIONS 5. Another famous identity that can be used to generate Pythagorean Triples is shown below:

2 222 2 2 22x y xy x y

The complicated side are shown on the diagram. (a) Prove this identity by expanding the products on both sides

of the equation.

(a) Prove this identity by expanding the product on the right-hand side of the equation.

(b) Use the identity to find the value of 3 310 9 without the use of your calculator. Show the steps in your calculation. Then, verify with your calculator.

2xy

2 2x y

2 2x y

(b) Generate the sides of the right triangle if 4x and 1y . Show that these sides satisfy the Pythagorean Theorem.

(c) Reasoning: This relationship leads to the conclusion that there is no Pythagorean triple that contains the integer 2. Why?

Name: ___________________________________ Date: _________________


INTRODUCTION TO RATIONAL FUNCTIONS COMMON CORE ALGEBRA II

Rational functions are simply the ratio of polynomial functions. They take on more interesting properties and have more interesting graphs than polynomials because of the interaction between the numerator and denominator of the fraction. In Common Core Algebra II, we will be primarily concerned with the algebra of these functions. But in this lesson we will explore some of their characteristics.

Exercise #1: Consider the rational function given by 6

3

xf x

x

.

Exercise #2: Find all values of x for which the rational function 2

5

2 11 6

xh x

x x

is undefined. Verify by

using your calculator to evaluate this expression for these values.

Exercise #3: Which of the following represents the domain of the function 2

3

6 16

xf x

x x

?

(1) | 4x x (3) | 2 and 8x x

(2) | 3x x (4) | 6 and 3x x

(a) Algebraically determine the y-intercept for this function.

(c) For what value of x is this function undefined? Why is it undefined at this value?

(b) Algebraically determine the x-intercept of this function. Hint – a fraction can only equal zero if its numerator is zero.

(d) Based on (c), state the domain of this function in set-builder notation.


Exercise #4: If 2 13 2 and

5

xg x x f x

x

then find:

(a) 2f g (b) 2f g (c) f g x

Exercise #5: Find formulas for the inverse of each of the following simple rational functions below. Recall that as a first step, switch the roles of x and y.

(a) 2

xy

x

(b)

3

2

xy

x

Exercise #6: The function 2 8

4

xf x

x

is either an even or an odd function. Determine which it is and justify.

Based on your answer, what type of symmetry must this function have? Use your calculator to sketch a graph to verify.

5

10 10

5

y

x

Name: ___________________________________ Date: _________________


INTRODUCTION TO RATIONAL FUNCTIONS COMMON CORE ALGEBRA II HOMEWORK

FLUENCY

1. Which of the following values of x is not in the domain of 3

7

xf x

x

?

(1) 7x (3) 3x

(2) 7x (4) 3x

2. Which of the following values of x is not in the domain of 4 1

2 1

xg x

x

?

(1) 1

2x (3)

1

4x

(2) 1x (4) 3x

3. Which values of x, when substituted into the function 2

4

2 8

xy

x x

, would make it undefined?

(1) 2 and 8x (3) 4 and 4x

(2) 4 and 8x (4) 4 and 0x

4. Which of the following represents the domain of 2

2

4

5 14

xy

x x

?

(1) | 2x x (3) | 4 and 14x x

(2) | 7 and 2x x (4) | 5 and 14x x

5. Which of the following represents the domain of 2

3 1

2 10

xg x

x x

?

(1) 1

|3

x x

(3) 1

| and 52

x x

(2) 1 1

| and 3 2

x x

(4) 5

| 2 and 2

x x

6. If 2 4

2 7 and 2 1

xf x x g x

x

then 5g f ?

(1) 1 (3) 6

(2) 2 (4) 3


7. If 3 2 and 4 1 then ?

2

xf x g x x f g x

x

(1) 7 3

2

x

x

(3)

12 5

8 2

x

x

(2) 12 9

8 2

x

x

(4) 5 4x

x

8. The y-intercept of the rational function 2 15

3

xy

x

is

(1) 15 (3) 3

(2) 5 (4) 12

9. Find formulas for the inverse of each of the following rational functions.

(a) 5

2

xy

x

(b)

3 2

4

xy

x

10. Consider the rational function 2

2

9

1

xy

x

.

(d) Is this an even or an odd function? Explain graphically.

y

x

2

5

10

5

(c) Sketch the function on the axes below. Clearly label your x and y intercepts.

(a) Find the function’s y-intercept algebraically.

(b) Find the function’s x-intercepts algebraically.

Name: ___________________________________ Date: _________________


SIMPLIFYING RATIONAL EXPRESSIONS COMMON CORE ALGEBRA II

Simplifying a rational expression into its lowest terms is an extremely useful skill. Its algebra is based on how we simply numerical fractions. The basic principle is developed in the first exercise. Exercise #1: Recall that to multiply fractions, one simply multiplies their numerators and denominators. Every time we simplify a fraction, we are essentially finding all common factors of the numerator and denominator and dividing them to be equal to one. Key in this process is that the numerator and denominator must be factored and only factors cancel each other. This is true whether our fraction contains monomial, binomial, or polynomial expressions.

Exercise #2: Simplify each of the following monomials dividing other monomials.

(a) 5 6

8 3

3

6

x y

x y (b)

10 8

2

20

4

x y

x (c)

3

5 8

7

21

x y

x y

Exercise #3: Which of the following is equivalent to 6 3

2 6

10

15

x y

x y?

(1) 3

2

2

3

x

y (3)

4

3

2

3

x

y

(2) 8

9

3

2

x

y (4)

2

3

3

2

x

y

(a) Simplify the numerical fraction 18

12 by first

expressing it as a product of two fractions, one of which is equal to one.

(b) Simplify the algebraic fraction 2 9

2 6

x

x

by first

expressing it as the product of two fractions (factor!), one of which is equal to one.


When simplifying rational expressions that are more complex, always factor first, then identify common factors that can be eliminated. Exercise #4: Simplify each of the following rational expressions.

(a) 2

2

5 14

4

x x

x

(b) 24 1

10 5

x

x

(c) 2

2

3 14 8

16

x x

x

A special type of simplifying occurs whenever expressions of the form and x y y x are involved.

Exercise #5: Simplify each of the following fractions.

(a) 9 6

6 9

(b) 15 3

3 15

(c) a b

b a

Exercise #6: Which of the following is equivalent to 2

2 10

25

x

x

?

(1) 2

5x

(3) 5

2

x

(2) 2

5

x (4)

2

5x

Exercise #7: Which of the following is equivalent to 2 6 9

18 6

x x

x

?

(1) 3

6

x (3)

3

9

x

(2) 3

6

x (4)

3

6

x

Name: ___________________________________ Date: _________________


SIMPLIFYING RATIONAL EXPRESSIONS COMMON CORE ALGEBRA II HOMEWORK

FLUENCY

1. Write each of the following ratios in simplest form.

(a) 8

2

5

20

x

x (b)

3

12

12

8

y

y

(c)

10 2

4 5

6

15

x y

x y (d)

3 7

6 10

24

12

x y

x y

2. Which of the following is equivalent to the expression 6 4

2 6

4

12

x y

x y?

(1) 4

23

x

y (3)

3

2

3x

y

(2) 2

3

3y

x (4)

3

23

x

y

3. Simplify each of the following rational expressions.

(a) 2 25

4 20

x

x

(b) 2

2

11 24

9

x x

x

(c) 2

2

4 1

5 10

x

x x

(d) 2

2

9 4

3 4 4

x

x x

(e) 2

2

7 42

2 48

x x

x x

(f) 2

2

2 3 5

25 4

x x

x


4. Which of the following is equivalent to the fraction 2

2

9 18

15 5

x x

x x

?

(1) 3

5

x

x

(3)

6

5

x

x

(2) 6

5

x

x

(4)

6

5

x

x

5. The rational expression 2

2

2 7 6

4

x x

x

can be equivalently rewritten as

(1) 2 3

2

x

x

(3) 2 3

2

x

x

(2) 2 1

6

x

x

(4) 3 2

2

x

x

6. Written in simplest form, the fraction 2 2

5 5

y x

x y

is equal to

(1) 5 5y x (3)

5

x y

(2) 5

y x (4)

5

x y

REASONING

7. When we simplify an algebraic fraction, we are producing equivalent expressions for most values of x.

Consider the expressions 2 4 2

and 2 4 2

x x

x

.

(c) Clearly these two expressions are not equivalent for

an input value of 2x . Explain why.

(a) Show by simplifying the first expression that these two are equivalent.

(b) Use your calculator to fill out the value for both of these expressions to show their equivalence.

x 2 4

2 4

x

x

2

2

x

0

1

2

3

4

Name: ___________________________________ Date: _________________


MULTIPLYING AND DIVIDING RATIONAL EXPRESSIONS COMMON CORE ALGEBRA II

Multiplication of rational expressions follows the same principles as those involved in simplifying them. The process is illustrated in Exercise #1 with both a numerical and algebraic fraction. Notice the parallels.

Exercise #1: Simplify each of the following rational expressions by factoring completely. For the numerical fraction, make sure to prime factor all numerators and denominators.

(a) 6 10

8 3 (b)

2 2

2 2

4 3 15

6 6 12

x x x

x x x x

The ability to “cross-cancel” with fractions is a result of the two facts: (1) to multiply fractions we multiply their respective numerators and denominators and (2) multiplication is commutative. The keys to multiplication, then, are the same as that for simplifying – factor and then reduce.

Exercise #2: Simplify each of the following products.

(a) 7 2

3 3

8 10

5 6

y x

x y (b)

2 3 2

5 2 5 7

6 10

4 9

x y xy

x y x y

(c) 2 2

2

2 12 4 12

4 8 36

x x x x

x x

(d)

2 2

3 2 2

9 4 4

2 6 2 3

x x x

x x x x


Division of rational expressions continues to follow from what you have seen in previous courses. Since division by a fraction can always be thought of in terms of multiplying by it reciprocal, these problems simply involve an additional step.

Exercise #3: Perform each of the following division problems. Express all answers in simplest form.

(a) 2 8

5 7

15 5

6 2

x x

y y (b)

6 2

3

30 24

20 8

y y

x x

(c) 2 22 8 16

8 16 2 10

x x x

x x

(d)

2

2 2

9 1 5 15

3 7 2 5 14

x x

x x x x

Exercise #4: When 2 25

3

x

x

is divided by

5

9

x

x

the result is

(1) 5

27

x

x

(3)

20

3

x

(2) 3 15x (4) 9 5x

Name: ___________________________________ Date: _________________


MULTIPLYING AND DIVIDING RATIONAL EXPRESSIONS COMMON CORE ALGEBRA II HOMEWORK

SKILLS

1. Express each of the following products in simplest form.

(a) 4

8 2

12 15

5 30

x y

y x (b)

2 3

9 6

14 10

15 21

a b

b a (c)

3 7 2

5 2 3

4 3 30

9 10 8

x y z

z x y

2. Write each of the following products in simplest form.

(a) 2

2

9 16 8 8

12 16 3 4

x x

x x x

(b)

2 2

2 2

12 2 15

8 15 16

x x x x

x x x

(c) 2 2

3 2 2

2 7 4 12 24

8 4 6 8

x x x x

x x x x

(d)

2 2

2 2

7 8 3 4 1

1 9 1

x x x x

x x


3. When 1024

2

x

y is divided by

2

8

36

6

x

y the result is

(1) 8 72x y (3) 8

73

x

y

(2) 5

7

3

2

x

y (4)

4

72

x

y

4. Express the result of each division problem below in simplest form.

(a) 3 2 2

2 2

5 10 5 6

10 40 12

x x x x

x x x x

(b)

2

2 2

24 18 2 2

9 16 3 7 4

x x x

x x x

(c) 2 2

4 3 3 2

6 8 4 1

3 6 2

x x x

x x x x

(d)

2 2

2

49 2 35

9 14 6 3

x x x

x x x

Name: ___________________________________ Date: _________________


COMBINING RATIONAL EXPRESSIONS WITH ADDITION AND SUBTRACTION COMMON CORE ALGEBRA II

Occasionally it will be important to be able to combine two or more rational expressions by addition. Keep in mind two key principles that dictate fraction addition. Exercise #1: Combine each of the following fractions by first finding a common denominator. Express your answers in simplest form.

(a) 2 5 4 2

4 6

x x

x x

(b)

4 1 5

5 10

x x

x

(c)

2

3 1

4 2x x

Each of the combinations in Exercise #1 should have been reasonably easy because each denominator was monomial in nature. If this is not the case, then it is wise to factor the denominators before trying to find a common denominator. Exercise #2: Combine each of the following fractions by factoring the denominators first. Then find a common denominator and add.

(a) 2

4 5

5 15 9y y

(b)

2 2

3 2

9 20 6 8

x

x x x x

TWO GUIDELINES FOR ADDITION AND SUBTRACTION OF FRACTIONS

1. Fractions must have a common denominator. 2. Denominators can only be changed by multiplying the overall fraction by one.


Subtraction of rational expressions is especially challenging because of errors that naturally arise when students forget to distribute the subtraction (or the multiplication by -1). Still, with careful execution, these problems are no different than their addition counterparts. Exercise #3: Perform each of the following subtraction problems. Express your answers in simplest form.

(a) 2 2

3 7 3

4 4

x x

x x

(b)

2

3 2

4 1 10 5

x

x x

(c) 2 2

6

4 8 20

x

x x x

(d)

2 2

2 8

5 4 12 32

x

x x x x

Exercise #4: Which of the following is equivalent to 1 1

1x x

?

(1) 1

x

x (3)

2

1

x x

(2) 2

1

x x (4)

2 1

x

x

Name: ___________________________________ Date: _________________


COMBINING RATIONAL EXPRESSIONS WITH ADDITION AND SUBTRACTION COMMON CORE ALGEBRA II HOMEWORK

FLUENCY

1. Combine each of the following using addition. Simply you result whenever possible.

(a) 3 1 2 5

6 9

x x (b)

1

10 15

x

x (c)

2

3 5

7 14x x

2. Combine and simplify each of the following. Note that each pair of fractions already has a common

denominator.

(a) 3 7 2 3

2 2

x x

x x

(b)

5 2 3 8

4 12 4 12

x x

x x

(c)

2 2

2 2

6 8 4 2

25 25

x x x x

x x

3. Combine each of the following using addition. Simplify your final answers.

(a) 2

2 3

5 25 3 40

x x

x x x

(b)

2 2

4 2

24 128 12 32

x

x x x x


4. Which of the following represents the sum of 1 1

and 1 1x x

?

(1) 2

2

1

x

x (3)

2

1x

(2) 1

x (4)

2

2

1

x

x

5. When the expressions 2

2 2

8 3 6 and

9 9

x x x

x x

are added the result can be written as

(1) 5

3

x

x

(3) 2

3

x

x

(2) 2

3

x

x

(4) 7

3

x

x

6. Express each of the following differences in simplest form.

(a) 2 2

2 4

4 32 16

x

x x x

(b)

2 2

2 3 3

8 6 1 2 1

x

x x x x

7. When 7 14

3 12

x

x

is subtracted from 2 6

3 12

x

x

the result can be simplified to

(1) 5

3 (3)

10

3

(2) 2

3 (4)

7

3

Name: ___________________________________ Date: _________________


COMPLEX FRACTIONS COMMON CORE ALGEBRA II

Complex fractions are simply defined as fractions that have other fractions within their numerators and/or denominators. To simplify these fractions means to remove these minor fractions and then eliminate any common factors. The key, as always, is to multiply by the number one in ways that simplify the fraction.

Exercise #1: Consider the complex fraction

1 19 18

13

.

By multiplying the major fraction by the number one, by using the least common denominator, we will always eliminate the minor fractions (by turning them into integer expressions). Exercise #2: Simplify each of the following complex fractions.

(a)

1 12 10

25

(b)

2 235 53

x

x

(c)

3 18 47 32 4

x

x

(a) What is the least common denominator amongst the three minor fractions?

(b) Multiply the numerator and denominator of the major fraction by your answer in part (a) and then simplify your result.

(c) Why is it acceptable to perform the operation in part (b)? What number are you effectively multiplying by?


These types of problems can certainly involve more complicated secondary simplification. Don’t forget the primary use of factoring in order to simplify.

Exercise #3: Simplify each of the following complex fractions.

(a) 2

2

1 223 3

2

x

x x

(b)

2

2 251 1

5

x

x x

(c)

1 212 6

412 3

xx

xx

If the denominators of the minor fractions become more complex, be sure to factor them first, just as you did with the addition and subtraction in the previous lesson.

Exercise #4: Simplify each of the following complex fractions.

(a)

2

4 22 4

12 242 8

x xx

x x

(b) 2

2

16 2

48 12

xx x

xx x

Name: ___________________________________ Date: _________________


COMPLEX FRACTIONS COMMON CORE ALGEBRA II HOMEWORK

FLUENCY

1. Simplify each of the following numerical complex fractions.

(a)

1 34 20

12

(b)

5 118 6

13

(c)

3 14 5

14

2. Simplify each of the following complex fractions.

(a)

1 12 33 1

10 5

x

x

(b)

12

25

1

x

x

(c)

2

1 18 21 1

12 3

x

x x

3. Simplify each of the following complex fractions.

(a) 2

2

5 531 33

x x

x

(b)

1 210 10

12 10

xx

x

(c)

2

33

41

28

x

x


4. Simplify each of the following numerical complex fractions.

(a)

2

44 105 10

14 40

xx x

xx x

(b) 2

2

3 2 81 4

2 125 4

xx x

x xx x

5. Which of the following is equivalent to

2

1 111

x x

x x

?

(1) 1 (3) 1

x

x

(2) 2

1x (4) 2x x

REASONING 6. Since one can multiply by the number 1 at any point in an expression, simplify the following complex

fraction by simplifying the more minor complex fraction first, then continue

2

1 12

1 110 1

21 1

10 5

x

xx

x x

Name: ____________________________________ Date: __________________


POLYNOMIAL LONG DIVISION COMMON CORE ALGEBRA II

We have worked to simplify rational expressions (polynomials divided by polynomials). In this lesson, we will look more closely at the division of two polynomials and how it is analogous to the division of two integers.

Exercise #1: Consider the division problem 1519 7 , which could also be written as 1519

7 and 7 1519 .

Exercise #2: Now let's see how this works out when we divide two polynomials.

(c) Write 22 15 20

6

x x

x

in the form 6

rq x

x

, where q x is a polynomial and r is a constant, by

performing polynomial long division. Also, write the result an equivalent multiplication equation.

(a) Find the result of this division using the standard long division algorithm. Is there a remainder in this division?

(b) Rewrite your result from (a) as an equivalent multiplication equation.

(c) Now evaluate 1522

7 using long division. Write

your answer in Ra b form and in b

ac

form,

(d) Write your answer from part (c) as an equivalent multiplication equation.

(a) Simplify 22 15 18

6

x x

x

by performing

polynomial long division.

(b) Rewrite the result you found in (a) as an equivalent multiplication equation.


So, when we divide two polynomials, we always get another polynomial and a remainder. This is known as writing the rational expression in quotient-remainder form.

Exercise #3: Write each of the following rational expressions in the form r

q xx a

form.

(a) 2 2 5

3

x x

x

(b) 22 23 20

10

x x

x

Sometimes we can use the structure of an expression instead of polynomial long division.

Exercise #4: Consider the expression 8

3

x

x

. We would like to write this as 5

ba

x

.

We can extend what we did in the last problem to more challenging structure problems.

Exercise #5: Write each of the following in the form of b

ax r

.

(a) 4 13

2

x

x

(b) 3 5

4

x

x

(a) Write the numerator as an equivalent expression involving the expression 3x .

(b) Use the fact that division distributes over addition to write the final answer.

Name: ____________________________________ Date: __________________


POLYNOMIAL LONG DIVISION COMMON CORE ALGEBRA II HOMEWORK

FLUENCY

1. Write each of the following rational expressions in the form r

ax b

. Do these by rewriting your

numerator as was done in Exercises #4 and #5.

(a) 6

2

x

x

(b) 10

3

x

x

(c) 2 5

2

x

x

(d) 5 2

4

x

x

2. If the expression 10 11

2 1

x

x

was placed in the form 52 1

a

x

, then which of the following would be the

value of a? (1) 6 (3) 3 (2) 7 (4) 5 3. Use polynomial long division to simplify each of the following ratios. There should be a zero remainder.

(a) 2 5 24

3

x x

x

(b) 26 11 10

3 2

x x

x


4. Use polynomial long division to write each of the following ratios in rq x

x a

form, where q x is a

polynomial and r is the remainder.

(a) 2 6 11

4

x x

x

(b) 2 2 25

7

x x

x

(c) 23 17 25

4

x x

x

(d) 25 41 3

8

x x

x

5. Write each of the following in rq x

x a

. The polynomial q x will now be a quadratic.

(a) 3 27 17 41

5

x x x

x

(b) 3 22 11 22 25

3

x x x

x

Name: ____________________________________ Date: __________________


THE REMAINDER THEOREM COMMON CORE ALGEBRA II

In the last lesson, we saw how two polynomials, when divided, resulted in another polynomial and a remainder. The remainder has a remarkable property in certain types of division. We will explore this relationship in the first exercise.

Exercise #1: Consider each of the following scenarios where we have p x

x a. In each case, simplify the division

using polynomial long division and then evaluate p a .

(a) 2 8 18

2

x x

x

2 8 18 2p x x x p

(b) 2 2 25

7

x x

x

2 2 25 7p x x x p

(c) 22 11 11

3

x x

x

22 11 11 7p x x x p

(d) 23 7 20

4

x x

x

23 7 20 4p x x x p


Exercise #2: If the ratio 2 11 22

9

x x

x

was placed in the form 9

rq x

x

, where q x is a linear function,

then which of the following is the value of r? (1) 3 (3) 9 (2) 5 (4) 4 In the past, the remainder theorem was used primarily to aid in evaluating polynomials. These days it is the primary justification for telling if a linear expression is a factor of a polynomial.

Exercise #3: By definition x a is a factor of p x if p x

q xx a

, where q x is another polynomial.

What must be true of the remainder, p a , for x a to be a factor of p x ? Explain.

Exercise #4: Determine if each of the following are factors of the listed polynomials by evaluating the polynomials.

Exercise #5: For what value of k will 4x be a factor of 2 52x kx ? Show how you arrived at your answer.

THE REMAINDER THEOREM

When the polynomial p x is divided by the linear factor x a then the remainder will always be p a .

In other words:

p x p a

q xx a x a

(a) Is 3x a factor of 2 11 24p x x x ? (b) Is 5x a factor of 22 9 2p x x x ?

(c) Is 5x a factor of 3 2 19 10p x x x x ? (d) Is 1x a factor of 3 27 11 3p x x x x

Name: ____________________________________ Date: __________________


THE REMAINDER THEOREM COMMON CORE ALGEBRA II HOMEWORK

FLUENCY

1. Which of the following is the remainder when the polynomial 2 5 3x x is divided by the binomial

8x ?

(1) 107 (3) 3 (2) 27 (4) 9

2. If the ratio 22 17 42

5

x x

x

is placed in the form 5

rq x

x

, where q x is a polynomial, then which of

the following is the correct value of r? (1) 3 (3) 18 (2) 177 (4) 7

3. When the polynomial p x was divided by the factor 7x the result was 11

7x

x

. Which of the

following is the value of 7p ?

(1) 8 (3) 15 (2) 7 (4) It does not exist

4. Which of the following binomials is a factor of the quadratic 24 35 24x x ? Try to do this without factoring but by using the Remainder Theorem.

(1) 6x (3) 8x (2) 4x (4) 2x

5. Which of the following linear expressions is a factor of the cubic polynomial 3 29 16 12x x x ? (1) 6x (3) 3x (2) 1x (4) 2x


6. Consider the cubic polynomial 3 2 46 80p x x x x .

(a) Using polynomial long division, write the ratio of

3

p x

x in quotient-remainder form, i.e. in the form

3

rq x

x

. Evaluate 3p . How does this help you check your quotient-remainder form?

(b) Evaluate 5p . What does this tell you about the binomial 5x ?

(c) If 5

p xq x

x

, then use polynomial long division to find an expression for the polynomial.

(d) Use your answer from (c) to completely factor the cubic polynomial p x . Besides 5x , what are its

other zeroes? 7. For the cubic 3 27 13 3x x x has only one rational zero, 3x . Use polynomial long division to show

that the remainder is zero when dividing the cubic by 3x . Then use the quadratic formula to find the other two (irrational) zeroes.

Name: ___________________________________ Date: _________________


SOLVING FRACTIONAL EQUATIONS COMMON CORE ALGEBRA II

Equations involving fractions or rational expressions arise frequently in mathematics. The key to working with them is to manipulate the equation, typically by multiplying both sides of it by some quantity, that eliminates the fractional nature of the equation. The most common form of this practice is “cross-multiplying.”

Exercise #1: Use the technique of cross multiplication to solve each of the following equations.

(a) 4 5 1

2 5

x x (b)

1 2

2 6

x

x x

Since this technique should be familiar to students at this point, we will move onto a less familiar method when more than two fractions are involved. The next exercise will illustrate the process.

Exercise #2: Consider the equation 1 9 3

2 4 4x x .

It is very important to note the similarities and differences between this technique and the one employed to simplify complex fractions. With complex fractions we multiplied by one in creative ways. Here we are multiplying both sides of an equation by a quantity that removes the fractional nature of the equation.

Exercise #3: Which of the following values of x solves: 4 2 31

6 10 15

x x ?

(1) 14x (3) 8x (2) 6x (4) 11x

(a) What is the least common denominator for all three fractions in this equations?

(b) Multiply both sides of this equation by the LCD to “clear” the equation of the denominators. Now, solve the resulting linear equation.


These equations can involve quadratic as well as root expressions. The key, though, remains the same – multiplying both sides of the equation by the same quantity. Exercise #4: Solve each of the following equations for all values of x.

(a) 2

1 1 1 2

10 5x x x (b)

2 2

1 3 1 1 1

2 4 2x x x x

Because fractional equations often involve denominators containing variables, it is important that we check to see if any solutions to the equation make it undefined. These represent further examples extraneous roots. Exercise #5: Solve and reject any extraneous roots.

(a) 2

1 18 9

5 8 15 3

x

x x x x

(b)

2

4 1 1

4 12 6 2

x

x x x x

Name: ___________________________________ Date: _________________


SOLVING FRACTIONAL EQUATIONS COMMON CORE ALGEBRA II HOMEWORK

FLUENCY

1. Solve each of the following fractional equations. After “clearing” the denominators you should have a linear equation to solve.

(a) 2 1 3

3 6 2

x x (b)

13 4 31

2 15 6x x (c)

5 13

2 2x

2. Solve each of the fractional equations for all value(s) of x.

(a) 12

8xx

(b) 2

3 1 1 1

4 2 2 3x x x

(c) 17 11 5 8

3 3

x

x x x

(d)

10 13 11

2 1 3

x

x


3. Solve the following equation for all values of x. Express your answers in simplest a bi form.

3

9 1

x x

x

4. Solve the following equation for all values of x. Be sure to check for extraneous roots.

11

11 11

x

x x

5. Solve each of the following equations. Be sure to check for extraneous roots.

(a) 2

1 2 2

5 6 11 30

x

x x x x

(b)

2

3 1 28

7 7

x

x x x x

Name: ___________________________________ Date: _________________


SOLVING RATIONAL INEQUALITIES COMMON CORE ALGEBRA II

We have already seen the solving of inequalities including quadratic expressions. Rational inequalities, those that include algebraic fractions with variables in both their numerator and denominator, are important and pose an interesting challenge compared with quadratics. The first exercise will illustrate the thinking involved in finding the solution set to a rational inequality.

Exercise #1: Consider the rational inequality 5

03

x

x

.

(c) Enter the ratio 5

3

x

x

in your calculator to help determine values of x that solve this inequality. Plot its

solution on a number line and state the answer in set-builder notation. The key to solving rational inequalities that are compared to zero is to find the values of x that make the numerator or denominator equal to zero. These are known as the critical values of the rational expression. The rational expression can only change signs at these critical values.

Exercise #2: Solve the rational inequality 2

2

3 40

6 9

x x

x x

for all values of x. Show your solution on a number

line and state its solution in either interval or set-builder notation.

(a) At what x-value is the ratio 5

3

x

x

equal to zero?

Is this value part of the solution set?

(b) At what x-value is the ratio 5

3

x

x

undefined? Is

this value part of the solution set?


If a single algebraic ratio is compared to zero, then the solution method is fairly straightforward. It becomes more difficult in there exists more than one ratio or if the ratio is being compared to a quantity other than zero. In both cases, it is important to algebraically manipulate the expression so that we are comparing it to zero.

Exercise #3: Solve the rational inequality 58

x

x

. Represent your answer using a number line and using set-

builder or interval notation. Some of these inequalities can test all of your key fraction abilities.

Exercise #4: Solve the rational inequality 1 2 3

3 2

x x

x x

. Represent your answer using a number line and

using any appropriate notation.

Name: ___________________________________ Date: _________________


SOLVING RATIONAL INEQUALITIES COMMON CORE ALGEBRA II HOMEWORK

FLUENCY Solve each of the following rational inequalities. Show your answers using a number line and an appropriate notation.

1. 10

05

x

x

2.

2 10

3

x

x

3. 2

2

40

20

x

x x

4.

2

2

6 160

6

x x

x x

5. 2

2

6 90

4 3 1

x x

x x

6.

2

2

12 360

4 4 1

x x

x x


For problems 7 through 9, solve each rational inequality by first comparing it to zero. Represent your answers on a number line and using appropriate notation.

7. 1

23

x

x

8. 2 2 4

4 3

x x

x

9. 2

1 1 3

2 2 4x x x

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