Fundamental Skills of Algebra Slide 3 / 224

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Algebra II

Fundamental Skills of

Algebra

www.njctl.org

2013-09-11

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Table of Contents

Solving Equations and Inequalities

Factoring

Exponents

Radicals

click on the topic to go to that section

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Goals and ObjectivesStudents will be able to solve a wide variety of equations and inequalities.


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Why do we need this?Life throws at us many different types of

problems. Being able to solve these problems makes us successful.

Developing logic skills that we can apply in different situations helps us solve

those problems with ease.

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Steps for solving equations:

1) Remove any parentheses.2) Collect like terms.3) Move all variables to one side.4) Move all constants to one side.5) Remove any coefficient.

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Remember...

· There are typically two "sides" to an equation. They are separated by the equals sign.

· An equation is like a balance. What you do to one side, must be done to the other side!

· Use opposite operations (+/-, x/÷) when moving variables and constants across the equals sign.

· "Collecting like terms" is just organizing and simplifying each side of the equation. Do not use opposites when collecting like terms.

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1 Try on your own. Use the rules from your notes to help you.


-3(2x + 4) - 8 = 2(x - 5) - 2

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The nice part about how equations work is that you can always check your answer. Plug in your result to the original equation even if your first answer was incorrect. What happens?

-3(2x + 4) - 8 = 2(x - 5) - 2 -3(2x + 4) - 8 = 2(x - 5) - 2Plug in incorrect answer: Plug in correct answer:

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More examples:

3m - 4 + 2m = 5 + 4m

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More examples:

-2(a + 3) - 2 = 3a - 15 - 8

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and a few more... Will all of our answers be nice, whole numbers?

-4y - 5 + 3y - 6 = 4 + 2y + 7

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and a few more... Will all of our answers be nice, whole numbers?

3(x - 4) + 5 = 1 + 7(x - 2)

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2 Solve for x: 2(x - 4) - 3(x + 5) = 2x - 8


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3 Find m: 5m - 4 - 2m = 4m - 12 + 6Solving Equations and Inequalities

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4 Solve the equation: -6(y + 2) = -7(y - 3)


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5 Find x: 5 - 2x - 4 - 4x = -3 - 5x + 2x - 5


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6 Solve for x: -4 - (x - 5) = 2 - (3x - 8)Solving Equations and Inequalities

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Try:


-2 - (m - 3) = -(m - 4) - 2

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Sometimes, the variable cancels out and you are left with a numerical statement. If that happens, you will have one of two possible answers.

True Statement = infinitely many solutionsFalse Statement = no solution

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For example:

-p + 3 = -p + 6

results in 3 ≠ 6

This is a false statement and

there is no solution.

b - 8 = b - 8

results in -8 = -8

This is a true statement and

there are infinitely many

solutions.


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7 What is the solution to the equation: 5 - 2(x + 3) = -2x + 4

A -4B 0C No solutionD Infinitely many solutions


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8 Find x: -2(x - 3) + 4 = -x + 10

A 10B 0C No solutionD Infinitely many solutions


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9 Solve: m - 6 - 3 - 2m = m - 4 - m - 5 - m



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10 Find y: -4y + 8 - y = y - 6 - 8

A 11/3B 0C No solutionD Infinitely many solutions


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11 Find the answer to the equation: -2(x - 4) + 5(x + 3) = 3x - 12 + 2



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Now for fractions...

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Try:


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12 Solve:


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13 Find x:


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14 Solve the equation:


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15 Find m:


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16 Solve the equation:


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Solving formulas for specific variables incorporates all of these rules. Try...

4pc = 2t - 9dm solve for d

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Try...this one is a bit tougher...

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17 Solve the following formula for C.

A

B

C

D

E


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18 Solve for h:

A

B

C

D

E


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19 Solve for h:

A

B

C

D

E


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20 Solve for w:

A

B

C

D

E


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21 Solve for m:

A

B

C

D

E


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22 Solve for x:

A

B

C

D

E


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Remember inequalities?

≥, ≤ >, <[ , ] ( , )

All symbols indicate solutions will include referenced points.

All symbols indicate solutions will not include referenced points.

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You solve inequalities the same way that you solve equations. The only difference is that you flip an inequality symbol if you multiply or divide by a negative number.

Solve: 3m + 4 - 5m < 3m + 6

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Solutions to equations are single points. Solutions to inequalities are regions of points.


Draw a graph to represent the solution to the last problem: m > -2/5

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Try solving and graphing the inequality...

2x - 6 + 3 ≥ 5x - 4

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One more, solve and graph the inequality...


3(x - 2) - 4(x + 3) ≤ -2(x - 4)

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23 Solve for x: 5(x - 3) + 2 -(2x - 4)

A

B

CDE


x ≤ 17/7

x ≥ 17/2

x ≤ 21/5x ≤ 14/5

x ≥ 21/5

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24 Solve the following inequality: 6y - 2 - 2y > 3y + 5

A y > 7/9B y < 7/9C y > 7/5D y > 3/5E y > 7


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25 Find values for p: 2(p + 3) - 5 4(p + 4)

A

B

C

D

E Answer not listed.


p ≥ -15/2

p ≤ -15/2

p ≥ -5/2

p ≤ -5/2

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26 Which of the following numbers would be a solution to the following inequality? 3p + 6 - 5p > 10p - 4 - 8

A 0B 5/2C 5D 15/4E 9


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27 Solve for n: 4 - 3(n - 3) 2(n + 4) - 2

A

B

C

D

E Answer not listed.

n ≤ -11/5

n ≥ -11/5n ≥ 19/5

n ≤ 19/5


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Factoring


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Factoring

Goals and ObjectivesStudents will be able to factor complex expressions and solve equations using factoring.

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Factoring

Why do we need this?The more we can simplify a problem, the

easier it is to solve. Factoring allows us to break up expressions into smaller parts

and, much of the time, simplify our strategies to solve them.

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Factoring

Multiply the following:

3x(2x2 + 5) (x + 3)(x - 9) (2x - 1)(3x - 4)

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Factoring

Factoring is undoing what you just did. It breaks up expressions into parts and pieces.

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Factoring

Factor out the GCF (greatest common factor).

8x + 32y 5a2b - 10ab 12a3b2 + 4ab2

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Factoring

You can easily check your answer by distributing the GCF back to all of your terms.

Factor then check your answer:

3x2y2 + 12xy2 + 6y2 6m4n3 + 18m3n2 - 36m2n

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28 What is the GCF of the following expression? 24a3b3c - 6ab3c

A 6a3b3c B 6abc C ab3c D 6ab3c E a3b3c

Factoring

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29 Factor:

A 3m5n2(4 - n - 5m)

B m5n2(12 - 3n - 15m)

C 3m6n3(4 - 5n)

D 3m4n2(4mn - 5m2)

E 3m4n2(4 - mn - 5m2)

Factoring

12m4n2 - 3m5n3 - 15m6n2

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30 Factor out the GCF:

A x2y2z3(x2 - xy + y2)

B xyz(x2 - xy + y2)

C x3y2z3(x - y - y2)

D x2y3z3(x2 - x - y)

E x2y2z3(x - xy + y2)

x4y2z3 - x3y3z3 + x2y4z3

Factoring

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31 Factor the following:

A -4x3y2(2x + 3y)

B -4x3y2(2x - 3y)

C -8x3y2(x + 4y)

D -8x3y2(x - 4y)

E -8x4y2(1 - 4y)

-8x4y2 - 12x3y3

Factoring

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32 Factor out the GCF: 15m3n - 25m2 - 15mn3

A 15m(mn - 10m - n3)

B 5m(3m2n - 5m - 3n3)

C 5mn(3m2 - 5m - 3n2)

D 5mn(3m2 - 5m - 3n)

E 15mn(mn - 10m - n3)

Factoring

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33 Factor:

A m3n3(2m - 2)

B -2mn(m3n + m)

C -2mn(m3n - m)

D -2m3n3(m2 + 1)

E -2m3n3(m2 - 1)

Factoring

-2m4n3 - 2m3n3

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34 Factor out the GCF:

A 14p3q7(1p - 14)

B 14p2q6(pq - 14)

C 14p2q6(pq - 2)

D p3q6(14q - 28)

E 14pq(p2q6 - 2pq5)

Factoring

14p3q7 - 28p2q6

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Factoring

Factoring quadratics of the form x2 + bx + c.

Try: x2 - x - 6

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Factoring

Here are some tips to help you factor:

Read the expression backwards:

x2 - x - 6

"factors of 6 that subtract to -1"

x2 + 6x + 8

"factors of 8 that add up to 6"

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Factoring

The signs in front of c will also help you factor:

x2 + bx + c ⇒ (x + m)(x + n)

x2 - bx + c ⇒ (x - m)(x - n)

x2 + bx - c

x2 - bx - c

If the sign in front of c is +: If the sign in front of c is -:

(x - m)(x + n)⇒

⇒

same signs, both +

same signs, both -

different signs

different signs

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Factoring

Here is the last tip...

Use a factor tree to help you find the factors of c. If none of the pairs add or subtract to the middle term, the quadratic is not factorable.

Factor trees: 12

>

1 122 63 4

24

>

1 242 123 84 6

48

>

1 482 243 164 126 8

72

>

1 722 363 244 186 128 9

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Factoring

x2 + 6x + 8

Read it backwards: "factors of 8 that add up to 6."

Use the pattern from c: "same signs, both +."

Now factor it!

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Factoring

Use the tips to help you factor the following quadratics.

x2 - 5x - 24 a2 - 13a + 30 m2 + 4m - 35

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Factoring

Try...x2 - 8xy + 16y2

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35 Factor the trinomial:

A (x - 7y)(x + 6y)

B (x + 7y)(x - 6y)

C (x - 42y)(x + y)

D (x + 42y)(x - y)

E Solution not shown

Factoring

x2 - xy - 42y2

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36 Factor:

A (x - 12)(x + 2)

B (x + 12)(x - 2)

C (x - 6)(x +4)

D (x - 6)(x - 4)


Factoring

x2 - 10x + 24

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37 Factor the quadratic:

A (x + 5)(x + 1)

B (x - 4)(x - 1)

C (x + 2)(x + 3)

D (x + 1)(x - 4)


x2 + 5x + 4

Factoring

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38 Factor:

A (x - 12)(x - 6)

B (x - 6)(x - 3)

C (x - 6)(x + 3)

D (x - 9)(x + 2)


x2 - 3x - 18

Factoring

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39 Factor:

A (x - 5)(x - 5)

B (x - 5)(x + 5)

C (x + 15)(x + 10)

D (x - 15)(x - 10)


Factoring

x2 + 10x + 25

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FactoringFactoring quadratics of the form ax2 + bx + c.

3x2 + 17x + 10

With just a few extra steps, you can factor quadratics with a leading coefficient (a) just like the previous ones.

Mulitiply a and c ⇒ x2 + 17x + 30Factor the result ⇒ (x + 15)(x + 2)

Divide the numbers by a ⇒ (x + 15)(x + 2) 3 3

Reduce any fractions ⇒ (x + 5)(x + 2)3

Move any remaining denominators ⇒ (x + 5)(3x + 2)

The answer is (x + 5)(3x + 2)

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Factoring

Try this one. Make sure you reduce the fractions where possible.

10x2 - 31x + 15

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Factoring

Here is another one to practice on...

3x2 + 5x - 2

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Factoring

Try... 4x2 + 4x +1

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40 Factor:

A (6x - 5)(x - 6) B (6x - 1)(x - 6) C (3x - 2)(2x + 3) D (3x + 2)(2x - 3) E Solution not shown

Factoring

6x2 - 5x - 6

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41 Factor the following:

A (2x - 1)(5x + 3) B (2x + 1)(5x + 3) C (10x - 1)(x + 3) D (10x - 1)(x - 3) E Solution not shown

10x2 - 11x + 3

Factoring

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42 Which is a factor of ?

A (4x + 5)B (2x + 3)C (12x + 5)D (3x + 5)E (4x + 2)

12x2 + 23x + 10

Factoring

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43 Which is a factor of the quadratic?

A (x - 7y)B (2x - 8y)C (2x - 6y)D (x - 8y)E (2x - y)

Factoring

2x2 - 23xy + 56y2

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44 Which of the following is a factor of ?

A (2x - 3y)B (5x - 3y)C (3x - 2y)D (3x + 5y)E (2x - 5y)

6x2 - 19xy + 15y2

Factoring

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45 Factor:

A (x - 4y)(4x + 2y)B (x - 8y)(4x + y)C (2x + 4y)(2x + 2y)D (4x + y)(x + 8y)E Solution not shown

Factoring

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Factoring

Multiply:

(2x + 3)(2x - 3) (x + 3)(x2 - 3x + 9)

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To factor the difference of squares, the difference of cubes and the sum of cubes, use the following formulas:

Factoring

a2 - b2 ⇒ (a - b)(a + b)

a3 - b3 ⇒ (a - b)(a2 + ab + b2)

a3 + b3 ⇒ (a + b)(a2 - ab + b2)

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Try...

Factoring

4p2 - q2 16m2 - 1 64p3 + y3

a2 - b2 ⇒ (a - b)(a + b)a3 - b3 ⇒ (a - b)(a2 + ab + b2)a3 + b3 ⇒ (a + b)(a2 - ab + b2)

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Factoring

a2 - b2 ⇒ (a - b)(a + b)a3 - b3 ⇒ (a - b)(a2 + ab + b2)a3 + b3 ⇒ (a + b)(a2 - ab + b2)Factor...

25x2 - 81y2 x3y3 + 1 8m3 - 125n3

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46 Factor:

A (11m - 10n)(11m + 10m) B (121m - n)(m + 100n) C (11m - n)(11m + 100n) D Not factorable E Solution not shown

121m2 + 100n2

Factoring

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47 Factor:

A (3m - 8np)(3m + 8np) B (3m + 4np)(3m2 - 12mnp + 4n2p2) C (3m + 4np)(9m2 - 12mnp + 16n2p2) D Not factorable E Solution not shown

Factoring

27m3 + 64n3p3

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48 Factor:

A (a - 5b)(a + 5b) B (a - 5b)(a2 + 5ab + 5b2) C (a + 5b)(a2 - 5ab + 25b2) D Not FactorableE Solution not shown

Factoring

a3 - 125b3

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49 Factor:

A (m + n)(m - n) B (m + n)(m + n) C (m - n)(m - n) D Not factorableE Solution not shown

Factoring

m2 - n2

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50 Factor:

A (d - 1)(d + 1) B (d - 1)(d2 + d + 1) C (d + 1)(d2 - d + 1) D Not factorableE Solution not shown

Factoring

d3 - 1

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51 Factor:

A (3m + 2n)(3m - 2n) B (3m + 2n)(9m2 - 6mn + 4n2) C (3m - 2n)(9m2 + 6mn + 4n2) D Not factorableE Solution not shown

Factoring

27m3 + 4n3

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52 Factor:

A (16x - 2y)(16x + 2y) B (6x - 2y)(6x2 + 12xy + 2y2) C (6x - 2y)(36x2 + 12xy + 4y2) D Not factorableE Solution not shown

Factoring

216x3 - 8y3

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Factoring

Factoring by Grouping.

What happens when there are 4 terms?

4ap - 4a + 3xp - 3x

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Factoring

Try... xy + 4x - 3y - 12

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Factoring

Two more...pq + 4p + 3q + 12 mn - pm - qn + qp

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53 Factor by grouping:

A (r + 4)(s - 7) B (r - 7)(s + 4) C (rs - 7)(rs + 4) D Not factorableE Solution not shown

Factoring

rs - 7r + 4s - 28

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54 Factor:

A (n - 3)(m + 4n) B (n - 3)(m - 4n) C (n + 4)(m - n) D Not factorableE Solution not shown

Factoring

mn + 3m - 4n2 - 12n

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55 Factor:

A (3k - 2)(g + 6) B (3g + 2)(k - 6) C (3k - 2)(g + 6) D Not factorableE Solution not shown

Factoring

3gk - 18g + 2k - 12

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56 Factor by grouping:

A (m - 4)(2p - 7) B (m + 7)(2p + 4) C (m - 4)(2p + 7) D Not factorableE Solution not shown

Factoring

2mp - 8p - 7m + 28

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57 Factor by grouping

A (x + 4)(2y + 3) B (x + 4)(2y - 3) C (x - 3)(2y + 4) D Not factorableE Solution not shown

2xy + 3x + 8y - 12

Factoring

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58 Factor:

A (3m - 5)(p + 2n) B (3m + 5)(p - 2n) C (3m - 2n)(p + 5) D Not factorableE Solution not shown

Factoring

3mp + 15m - 2np - 10n

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Now, let's combine all of the situations. In any factoring problem, factor out the GCF first.

Factoring

Factor these completely...

2x3 - 22x2 + 48x 3m3n + 3m2n - 18mn

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Factoring

Factor completely...

4x3 - 32y3 54a4 + 2ab3

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59 Factor completely:

A -3mn(2m - 1)(m - 3) B 3mn(2m + 1)(m - 3) C -3n(2m - 1)(m2 + 3m + 9) D Not factorableE Solution not shown

Factoring

-6m3n + 21m2n - 9mn

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A -3(2p + 5)(4p3 - 10p + 25) B -3p(16p2 + 25) C -3p(4p - 5)(4p - 5) D Not factorableE Solution not shown

Factoring

-48p3 + 75p

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A (4p2 - 3)(m - 4) B 4p2(pm - 4)(pm - 3) C 4p2m(m - 4)(p + 3) D Not factorableE Solution not shown

Factoring

4p3m - 12p3 - 16mp2 + 48p2

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A 2xy(9x - 1) B 2xy(3x - 1)(3x + 1) C 2y(9x2 - x) D Not factorableE Solution not shown

Factoring

18x3y - 2xy

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A -8ab(a2 + 4) B -4ab(2a2 + a + 3)C -4ab(2a + 3)(a - 1)D Not factorableE Solution not shown

Factoring

-8a3b - 4a2b + 12ab

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A 2b(4b - 1)(3b + 2) B 4b(6b2 + 6b - 1) C 2b(4b + 1)(3b - 2) D Not factorableE Solution not shown

24b3 + 10b2 - 4b

Factoring

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Factoring is often use to solve equations that are in polynomial form.

Factoring

Steps: 1) Move all terms to one side of the equation. (the other side becomes zero) 2) Factor the resulting polynomial. 3) Set each factor equal to zero. 4) Solve each equation. 5) Write the answers clearly.

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Factoring

Solve the equation by factoring:

x2 = 9x - 18

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One more...

Factoring

6x3 + 10x2 = 4x

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Another example...

Factoring

6m2 = 9m - 24m3

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65 Which of the following are solutions to the equation?

A 0B -1C -3/2D 3/2E -4F 4G -4/3H 4/3

Factoring

16x3 - 36x = 0

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66 Find all of the solutions to:

A 0B -1C 1D -3E 3F -4G 4H 12

Factoring

-3m3 + 3m2 = -36m

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67 Solve for p:

A 0B -1/2C 1/2D -2/5E 2/5F -4/3G -5/2H 5/2

Factoring

29p2 + 10p = -10p3

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68 Find the values for n:

A 0B -1/2C 1/2D -1/3E -2/3F 2/3G -4H 4

Factoring

18n4 + 48n2 = 84n3

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69 Find x:

A 0B -1/4C 1/4D -1/3E 1/3F -1/2G 1/2H 3

Factoring

6x4 = 5x3 - x2

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70 Solve by factoring:

A 0B -1/4C 1/4D -1/2E 1/2F 1G -2H 2

Factoring

p2 + p = 2p3

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Exponents


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Exponents

Goals and ObjectivesStudents will be able to simplify complex expressions containing exponents.

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Exponents

Why do we need this?Exponents allow us to condense bigger

expressions into smaller ones. Combining all properties of powers

together, we can easily take a complicated expression and make it

simpler.

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Exponents

Rules for working with exponents:

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Exponents

Multiplying powers of the same base:

(x4y3)(x3y)

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Exponents

(-3a3b2)(2a4b3)

Simplify:

(-4p2q4n)(3p3q3n)

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Work out:

Exponents

xy3 x5y4. (3x2y3)(2x3y)

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71 Simplify:

A m4n3p2 B m5n4p3 C mnp9 D Solution not shown

(m4np)(mn3p2)

Exponents

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72 Simplify:

A x4y5 B 7x3y5 C -12x3y4 D Solution not shown

Exponents

(-3x3y)(4xy4)

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73 Work out:

A 6p2q4 B 6p4q7 C 8p4q12 D Solution not shown

Exponents

2p2q3 4p2q4.

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74 Simplify:

A 50m6q8 B 15m6q8 C 50m8q15 D Solution not shown

Exponents

.5m2q3 10m4q5

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75 Simplify:

A a4b11 B -36a5b11 C -36a4b30 D Solution not shown

(-6a4b5)(6ab6)

Exponents

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Exponents

Dividing powers with the same base:

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Exponents

Simplify:

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Exponents

Try...

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77 Simplify:

A

B

C

D Solution not shown

Exponents

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79 Divide:

A

B

C


Exponents

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Exponents

Power to a power:

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Exponents

Simplify:

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Try:

Exponents

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81 Work out:

A

B

C


Exponents

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82 Work out:

A

B

C


Exponents

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83 Simplify:

A

B

C


Exponents

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84 Simplify:

A

B

C


Exponents

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85 Simplify:

A

B

C


Exponents

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Negative and zero exponents:Exponents

Why is this? Work out the following:

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Exponents

Sometimes it is more appropriate to leave answers with positive exponents, and other times, it is better to leave answers without

fractions. You need to be able to translate expressions into either form.

Write with positive exponents: Write without a fraction:

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Exponents

Simplify and write the answer in both forms.

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Exponents

Simplify and write the answer in both forms.

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Exponents

Simplify:

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Exponents

Write the answer with positive exponents.

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87 Simplify. The answer may be in either form.

A

B

C


Exponents

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88 Write with positive exponents:

A

B

C


Exponents

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89 Simplify and write with positive exponents:

A

B

C


Exponents

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90 Simplify. Write the answer with positive exponents.

A

B

C


Exponents

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91 Simplify. Write the answer without a fraction.

A

B

C


Exponents

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CombinationsExponents

Usually, there are multiple rules needed to simplify problems with exponents. Try this one. Leave your answers with positive exponents.

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Exponents

When fractions are to a negative power, a short cut is to flip the fraction and make the exponent positive.

Try...

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Exponents

Two more examples. Leave your answers with positive exponents.

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94 Simplify and write with positive exponents:

A

B

C


Exponents

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95 Simplify and write without a fraction:

A

B

C


Exponents

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96 Simplify. Answer may be in any form.

A

B

C


Exponents

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97 Simplify. Answer may be in any form.

A

B

C


Exponents

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Radicals


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page2svg

Goals and Objectives

Radicals

Students will be able to put problems in simplest radical form, as well as be able to add, subtract, multiply and divide radical expressions.

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Radicals

Why do we need this?Engineers and Scientists perform complicated

calculations. If a problem requires multiple operations and answers are rounded and reused for the next step,

what happens to the answer in the end? It is not as accurate as possible. In science, we round only at the end. This way, we can have more accurate answers.

Being able to work with radicals, keeping them in simplest radical form, will allow us to maintain exact

numbers as we work with problems.

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Radicals

= 4.898979485566356....

Since 24 is not a perfect square, you must round the decimal to make the number reasonable. Your answer is then not an exact number. Simplest radical form allows us to simplify

radicals, but keeps them as an exact number.

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Radicals

Putting radicals in simplest radical form.

Perfect square numbers {4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169...}

Use these numbers to factor the radical. Then simplify.

Try...

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98 Find:

Radicals

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100 Put in simplist radical form:

A

BCD Solution not shown

Radicals

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101 Simplify:

A

B

C


Radicals

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102 Simplify:

A

B

C


Radicals

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104 Simplify:

A

B

C


Radicals

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Simplify: 3x + 4x and 3x + 4y

Think about: and

Adding and subtracting radicals - relate it to what you know...Radicals

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Adding and subtracting radicals is the same as adding and subtracting terms with variables. If the roots do not match, you

cannot put them together.

Radicals

Try...

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Radicals

Can you simplify these?

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Radicals

Try...

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106 Simplify:

A

B

C


Radicals

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107 Put in simplest radical form and collect like terms:

A

B

C


Radicals

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Radicals

Multiplying Radicals

Rule: Radical times radical, whole number times whole number.

Work out:

**All answers must be left in simplest radical form.

. .

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Radicals

*Remember to leave your answers in simplest radical form

. .

Try...

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And...

Radicals

. .

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112 Work out:

A

B

C


Radicals

.

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113 Multiply:

A

B

C


Radicals

.

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114 Multiply:

A

B

C


Radicals

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116 Simplify:

A

B

C


Radicals

.

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Radicals

Dividing Radicals

Rules are the same for dividing radicals as with multiplying radicals: Radical divided by radical, whole number divided by whole number.

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Radicals

Dividing radicals is a bit tougher because each problem is different. You know you are done with a question when there is no radical in the denominator and any fraction is reduced.

Removing a radical from the denominator is called "rationalizing the denominator."

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Radicals

Again, each division problem is different. Try simplifying everything you can before rationalizing the denominator.

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Radicals

Variables work the same way. Try...

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117 Divide:

A

B

C


Radicals

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118 Simplify:

A

B

C


Radicals

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119 Rationalize the denominator and simplify:

A

B

C


Radicals

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120 Rationalize the denominator:

A

B

C


Radicals

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121 Simplify:

A

B

C


Radicals

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Fundamental Skills of Algebra Slide 3 / 224

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