Review 10.1-10.4 Polynomials

Review 10.1-Review 10.1-10.410.4

PolynomialsPolynomials

• Monomials - a number, a variable, or a product of a number and one or more variables. 4x, 20x2yw3, -3, a2b3, and 3yz are all monomials.

• Polynomials – one or more monomials added or subtracted

• 4x + 6x2, 20xy - 4, and 3a2 - 5a + 4 are all polynomials.

Vocabulary

Like TermsLike Terms

Like Terms refers to monomials that have the same variable(s) but may have different coefficients. The variables in the terms must have the same powers.

Which terms are like? 3a2b, 4ab2, 3ab, -5ab2

4ab2 and -5ab2 are like.

Even though the others have the same variables, the exponents are not the same.

3a2b = 3aab, which is different from 4ab2 = 4abb.

Like TermsLike Terms

Constants are like terms.

Which terms are like? 2x, -3, 5b, 0

-3 and 0 are like.

Which terms are like? 3x, 2x2, 4, x

3x and x are like.

Which terms are like? 2wx, w, 3x, 4xw

2wx and 4xw are like.

A polynomial with only one term is called a monomial. A polynomial with two terms is

called a binomial. A polynomial with three terms is called a trinomial. Identify the

following polynomials:

Classifying Polynomials

Polynomial DegreeClassified by degree

Classified by number of

terms6

–2 x

3x + 1

–x 2 + 2 x – 5

4x 3 – 8x

2 x 4 – 7x

3 – 5x + 1

0

1

1

4

2

3

constant

linear

linear

quartic

quadratic

cubic

monomial

monomial

binomial

polynomial

trinomial

binomial

Add: (x2 + 3x + 1) + (4x2 +5)

Step 1: Underline like terms:

Step 2: Add the coefficients of like terms, do not change the powers of the variables:

Adding PolynomialsAdding Polynomials

(x2 + 3x + 1) + (4x2 +5)

Notice: ‘3x’ doesn’t have a like term.

(x2 + 4x2) + 3x + (1 + 5)

5x2 + 3x + 6

Some people prefer to add polynomials by stacking them. If you choose to do this, be sure to line up the like terms!


(x2 + 3x + 1) + (4x2 +5)

5x2 + 3x + 6

(x2 + 3x + 1)

+ (4x2 +5)

Stack and add these polynomials: (2a2+3ab+4b2) + (7a2+ab+-2b2)

(2a2+3ab+4b2) + (7a2+ab+-2b2)(2a2 + 3ab + 4b2)

+ (7a2 + ab + -2b2)

9a2 + 4ab + 2b2


1) 3x3 −7x( ) + 3x3 +4x( ) = 6x3 −3x

2) 2w2 +w−5( ) + 4w2 +7w+1( )= 6w2 +8w−4

3) 2a3 +3a2 +5a( )+ a3 +4a+3( ) =

3a3 +3a2 +9a+3

• Add the following polynomials; you may stack them if you prefer:

Subtract: (3x2 + 2x + 7) - (x2 + x + 4)

Subtracting PolynomialsSubtracting Polynomials

Step 1: Change subtraction to addition (Keep-Change-Change.).

Step 2: Underline OR line up the like terms and add.

(3x2 + 2x + 7) + (- x2 + - x + - 4)

(3x2 + 2x + 7)

+ (- x2 + - x + - 4)

2x2 + x + 3

Subtracting PolynomialsSubtracting Polynomials

1) x2 −x−4( )− 3x2 −4x+1( )=−2x2 +3x−5

2) 9y2 −3y+1( )− 2y2 +y−9( )= 7y2 −4y+10

3) 2g2 +g−9( )− g3 +3g2 +3( )= −g3 −g2 +g−12

• Subtract the following polynomials by changing to addition (Keep-Change-Change.), then add:

1. Add the following polynomials:(9y - 7x + 15a) + (-3y + 8x - 8a)

Group your like terms.

9y - 3y - 7x + 8x + 15a - 8a

6y + x + 7a

Combine your like terms.

3a2 + 3ab + 4ab - b2 + 6b2

3a2 + 7ab + 5b2

2. Add the following polynomials:(3a2 + 3ab - b2) + (4ab + 6b2)

Add the polynomials.

+X2

11XX

XYYY

YY

1 11

XYY

Y 111

1. x2 + 3x + 7y + xy + 8

2. x2 + 4y + 2x + 3

3. 3x + 7y + 8

4. x2 + 11xy + 8

Line up your like terms. 4x2 - 2xy + 3y2

+ -3x2 - xy + 2y2

_________________________

x2 - 3xy + 5y2

3. Add the following polynomials using column form:

(4x2 - 2xy + 3y2) + (-3x2 - xy + 2y2)

Rewrite subtraction as adding the opposite.

(9y - 7x + 15a) + (+ 3y - 8x + 8a)

Group the like terms.

9y + 3y - 7x - 8x + 15a + 8a

12y - 15x + 23a

4. Subtract the following polynomials:(9y - 7x + 15a) - (-3y + 8x - 8a)

Rewrite subtraction as adding the opposite.

(7a - 10b) + (- 3a - 4b)Group the like terms.

7a - 3a - 10b - 4b4a - 14b

5. Subtract the following polynomials:(7a - 10b) - (3a + 4b)

Line up your like terms and add the opposite.

4x2 - 2xy + 3y2

+ (+ 3x2 + xy - 2y2)--------------------------------------

7x2 - xy + y2

6. Subtract the following polynomials using column form:

(4x2 - 2xy + 3y2) - (-3x2 - xy + 2y2)

Find the sum or difference.(5a – 3b) + (2a + 6b)

1. 3a – 9b

2. 3a + 3b

3. 7a + 3b

4. 7a – 3b

Find the sum or difference.(5a – 3b) – (2a + 6b)

1. 3a – 9b

2. 3a + 3b

3. 7a + 3b

4. 7a – 9b

Find the sum. Write the answer in standard format.

(5x 3 – x + 2 x

2 + 7) + (3x 2 + 7 – 4 x) + (4x

2 – 8 – x 3)

Adding Polynomials

SOLUTI

ON Vertical format: Write each expression in standard form. Align like terms.

5x 3 + 2 x

2 – x + 7

3x 2 – 4 x + 7

– x 3 + 4x

2 – 8+

4x 3 + 9x

2 – 5x + 6

Find the sum. Write the answer in standard format.

(2 x 2 + x – 5) + (x + x

2 + 6)

Adding Polynomials

SOLUTI

ON Horizontal format: Add like terms.

(2 x 2 + x – 5) + (x + x

2 + 6) =(2 x 2 + x

2) + (x + x) + (–5 + 6)

=3x 2 + 2 x + 1

Find the difference.

(–2 x 3 + 5x

2 – x + 8) – (–2 x 2 + 3x – 4)

Subtracting Polynomials

SOLUTION

Use a vertical format. To subtract, you add the opposite. This means you multiply each term in the subtracted polynomial by –1 and add.

–2 x 3 + 5x

2 – x + 8

–2 x 3 + 3x – 4– Add the opposite

No change –2 x 3 + 5x

2 – x + 8

2 x 3 – 3x + 4+


(–2 x 3 + 5x

2 – x + 8) – (–2 x 2 + 3x – 4)


SOLUTION

Use a vertical format. To subtract, you add the opposite. This means you multiply each term in the subtracted polynomial by –1 and add.

–2 x 3 + 5x

2 – x + 8

–2 x 3 + 3x – 4–

5x 2 – 4x + 12

–2 x 3 + 5x

2 – x + 8

2 x 3 – 3x + 4+


(3x 2 – 5x + 3) – (2 x

2 – x – 4)


SOLUTION

Use a horizontal format.

(3x 2 – 5x + 3) – (2 x

2 – x – 4)= (3x 2 – 5x + 3) + (–1)(2 x

2 – x – 4)

= x 2 – 4x + 7

= (3x 2 – 5x + 3) – 2 x

2 + x + 4

= (3x 2 – 2 x

2) + (– 5x + x) + (3 + 4)

MultiplyingMultiplyingPolynomialsPolynomials

Distribute and FOIL

Polynomials * Polynomials Polynomials * Polynomials

Multiplying a Polynomial by another Polynomial requires more than one distributing step.

Multiply: (2a + 7b)(3a + 5b)

Distribute 2a(3a + 5b) and distribute 7b(3a + 5b):

6a2 + 10ab 21ab + 35b2

Then add those products, adding like terms:

6a2 + 10ab + 21ab + 35b2 = 6a2 + 31ab + 35b2


An alternative is to stack the polynomials and do long multiplication.

(2a + 7b)(3a + 5b)

6a2 + 10ab21ab + 35b2

(2a + 7b)

x (3a + 5b)

Multiply by 5b, then by 3a:(2a + 7b)

x (3a + 5b)When multiplying by 3a, line up the first term under 3a.

+

Add like terms: 6a2 + 31ab + 35b2


Multiply the following polynomials:

1) x+5( ) 2x−1( )

2) 3w−2( ) 2w−5( )

3) 2a2 +a−1( ) 2a2 +1( )

Polynomials * Polynomials Polynomials * Polynomials 1) x+5( ) 2x−1( ) (x + 5)

x (2x + -1)

-x + -5

2x2 + 10x+

2x2 + 9x + -5

2) 3w−2( ) 2w−5( )(3w + -2)

x (2w + -5)-15w + 10

6w2 + -4w+

6w2 + -19w + 10


3) 2a2 +a−1( ) 2a2 +1( )

(2a2 + a + -1)

x (2a2 + 1)

2a2 + a + -1

4a4 + 2a3 + -2a2+

4a4 + 2a3 + a + -1

Types of PolynomialsTypes of Polynomials

• We have names to classify polynomials based on how many terms they have:

Monomial: a polynomial with one term

Binomial: a polynomial with two terms

Trinomial: a polynomial with three terms

F : Multiply the First term in each binomial. 2x • 4x = 8x2

There is an acronym to help us remember how to multiply two binomials without stacking them.

F.O.I.L.F.O.I.L.

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

(2x + -3)(4x + 5) = 8x2 + 10x + -12x + -15 = 8x2 + -2x + -15

O : Multiply the Outer terms in the binomials. 2x • 5 = 10x

I : Multiply the Inner terms in the binomials. -3 • 4x = -12x

L : Multiply the Last term in each binomial. -3 • 5 = -15

Use the FOIL method to multiply these binomials:

F.O.I.L.F.O.I.L.

1) (3a + 4)(2a + 1)

2) (x + 4)(x - 5)

3) (x + 5)(x - 5)

4) (c - 3)(2c - 5)

5) (2w + 3)(2w - 3)

Use the FOIL method to multiply these binomials:

F.O.I.L.F.O.I.L.

1) (3a + 4)(2a + 1) = 6a2 + 3a + 8a + 4 = 6a2 + 11a + 4

2) (x + 4)(x - 5) = x2 + -5x + 4x + -20 = x2 + -1x + -20

3) (x + 5)(x - 5) = x2 + -5x + 5x + -25 = x2 + -25

4) (c - 3)(2c - 5) = 2c2 + -5c + -6c + 15 = 2c2 + -11c + 15

5) (2w + 3)(2w - 3) = 4w2 + -6w + 6w + -9 = 4w2 + -9

There are three techniques you can use for multiplying polynomials.

The best part about it is that they are all the same! Huh? Whaddaya mean?

It’s all about how you write it…Here they are!1)Distributive Property2)FOIL3)Box Method

Sit back, relax (but make sure to write this down), and I’ll show ya!

1) Multiply. (2x + 3)(5x + 8)

Using the distributive property, multiply 2x(5x + 8) + 3(5x + 8).

10x2 + 16x + 15x + 24

Combine like terms.

10x2 + 31x + 24

A shortcut of the distributive property is called the FOIL method.

The FOIL method is ONLY used when you multiply 2 binomials. It is an

acronym and tells you which terms to multiply.

2) Use the FOIL method to multiply the following binomials:

(y + 3)(y + 7).

(y + 3)(y + 7). F tells you to multiply the FIRST

terms of each binomial.

y2

(y + 3)(y + 7). O tells you to multiply the OUTER


y2 + 7y

(y + 3)(y + 7). I tells you to multiply the INNER


y2 + 7y + 3y

(y + 3)(y + 7). L tells you to multiply the LAST

terms of each binomial.y2 + 7y + 3y + 21

Combine like terms.

y2 + 10y + 21

Remember, FOIL reminds you to multiply the:

First terms

Outer terms

Inner terms

Last terms

The third method is the Box Method. This method works for every problem!

Here’s how you do it. Multiply (3x – 5)(5x + 2)

Draw a box. Write a polynomial on the top and side of a box. It does not matter which goes where.

This will be modeled in the next problem along with

FOIL.

3x -5

5x

+2

3) Multiply (3x - 5)(5x + 2)

First terms:

Outer terms:

Inner terms:

Last terms:

Combine like terms.

15x2 - 19x – 10

3x -5

5x

+2

15x2

+6x

-25x

-10

You have 3 techniques. Pick the one you like the best!

15x2

+6x-25x-10

4) Multiply (7p - 2)(3p - 4)

First terms:

Outer terms:

Inner terms:

Last terms:

Combine like terms.

21p2 – 34p + 8

7p -2

3p

-4

21p2

-28p

-6p

+8

21p2

-28p-6p+8

Multiply (y + 4)(y – 3)1. y2 + y – 12

2. y2 – y – 12

3. y2 + 7y – 12

4. y2 – 7y – 12

5. y2 + y + 12

6. y2 – y + 12

7. y2 + 7y + 12

8. y2 – 7y + 12

Multiply (2a – 3b)(2a + 4b)1. 4a2 + 14ab – 12b2

2. 4a2 – 14ab – 12b2

3. 4a2 + 8ab – 6ba – 12b2

4. 4a2 + 2ab – 12b2

5. 4a2 – 2ab – 12b2

5) Multiply (2x - 5)(x2 - 5x + 4)You cannot use FOIL because they are not BOTH binomials. You must use the

distributive property.

2x(x2 - 5x + 4) - 5(x2 - 5x + 4)

2x3 - 10x2 + 8x - 5x2 + 25x - 20

Group and combine like terms.

2x3 - 10x2 - 5x2 + 8x + 25x - 20

2x3 - 15x2 + 33x - 20

x2 -5x +4

2x

-5

5) Multiply (2x - 5)(x2 - 5x + 4) You cannot use FOIL because they are not BOTH

binomials. You must use the distributive property or box method.

2x3

-5x2

-10x2

+25x

+8x

-20

Almost done!Go to

the next slide!

x2 -5x +4

2x

-5

5) Multiply (2x - 5)(x2 - 5x + 4) Combine like terms!

2x3

-5x2

-10x2

+25x

+8x

-20

2x3 – 15x2 + 33x - 20

Multiply (2p + 1)(p2 – 3p + 4)1. 2p3 + 2p3 + p + 4

2. y2 – y – 12

3. y2 + 7y – 12

4. y2 – 7y – 12

Example: (x – 6)(2x + 1)

x(2x) + x(1) – (6)2x – 6(1)

2x2 + x – 12x – 6

2x2 – 11x – 6

2x2(3xy + 7x – 2y)

2x2(3xy) + 2x2(7x) + 2x2(–2y)

2x2(3xy + 7x – 2y)

6x3y + 14x2 – 4x2y

(x + 4)(x – 3)

(x + 4)(x – 3)

x(x) + x(–3) + 4(x) + 4(–3)

x2 – 3x + 4x – 12

x2 + x – 12

(2y – 3x)(y – 2)

(2y – 3x)(y – 2)

2y(y) + 2y(–2) + (–3x)(y) + (–3x)(–2)

2y2 – 4y – 3xy + 6x

There are formulas (shortcuts) that work for certain polynomial

multiplication problems.

(a + b)2 = a2 + 2ab + b2

(a - b)2 = a2 – 2ab + b2

(a - b)(a + b) = a2 - b2

Being able to use these formulas will help you in the future when you have to factor. If you do not remember the formulas, you can always multiply

using distributive, FOIL, or the box method.

Let’s try one!1) Multiply: (x + 4)2

You can multiply this by rewriting this as (x + 4)(x + 4)

ORYou can use the following rule as a shortcut:

(a + b)2 = a2 + 2ab + b2

For comparison, I’ll show you both ways.

1) Multiply (x + 4)(x + 4)

First terms:

Outer terms:

Inner terms:

Last terms:

Combine like terms.

x2 +8x + 16

x +4

x

+4

x2

+4x

+4x

+16

Now let’s do it with the shortcut!

x2

+4x+4x+16

Notice you have two of

the same answer?

1) Multiply: (x + 4)2

using (a + b)2 = a2 + 2ab + b2

a is the first term, b is the second term(x + 4)2

a = x and b = 4Plug into the formula

a2 + 2ab + b2

(x)2 + 2(x)(4) + (4)2

Simplify.x2 + 8x+ 16

This is the same answer!

That’s why the 2 is in

the formula!

2) Multiply: (3x + 2y)2

using (a + b)2 = a2 + 2ab + b2

(3x + 2y)2

a = 3x and b = 2y

Plug into the formulaa2 + 2ab + b2

(3x)2 + 2(3x)(2y) + (2y)2Simplify

9x2 + 12xy +4y2

Multiply (2a + 3)2

1. 4a2 – 9

2. 4a2 + 9

3. 4a2 + 36a + 9

4. 4a2 + 12a + 9

Multiply: (x – 5)2

using (a – b)2 = a2 – 2ab + b2

Everything is the same except the signs!

(x)2 – 2(x)(5) + (5)2

x2 – 10x + 25

4) Multiply: (4x – y)2

(4x)2 – 2(4x)(y) + (y)2

16x2 – 8xy + y2

Multiply (x – y)2

1. x2 + 2xy + y2

2. x2 – 2xy + y2

3. x2 + y2

4. x2 – y2

5) Multiply (x – 3)(x + 3)

First terms:

Outer terms:

Inner terms:

Last terms:

Combine like terms.

x2 – 9

x -3

x

+3

x2

+3x

-3x

-9

This is called the difference of squares.

x2

+3x-3x-9

Notice the middle terms

eliminate each other!

5) Multiply (x – 3)(x + 3) using (a – b)(a + b) = a2 – b2

You can only use this rule when the binomials are exactly the same except for the sign.

(x – 3)(x + 3)

a = x and b = 3

(x)2 – (3)2

x2 – 9

6) Multiply: (y – 2)(y + 2)(y)2 – (2)2

y2 – 4

7) Multiply: (5a + 6b)(5a – 6b)

(5a)2 – (6b)2

25a2 – 36b2

Multiply (4m – 3n)(4m + 3n)

1. 16m2 – 9n2

2. 16m2 + 9n2

3. 16m2 – 24mn - 9n2

4. 16m2 + 24mn + 9n2

Simplify.1)

2)

2(x 5)

2(m 2)

(x 5)(x 5) 2x 10x 25

(m 2)(m 2) 2m 4m 4

Follow the pattern!2(a b) 2 2 a 2ab b 2(x 5) 2 x 10x 25 2(y 3) 2 y 6y 9

LastTerm

Twice the LastTerm

Square of the Last Term

Difference of Squares.

Multiply.

1)

2)

3)

4)

(x 3)(x 3)

(m 7)(m 7)

(y 10)(y 10)

(t 8)(t 8)

2x 9 2m 49 2y 100

2t 64

Inner and Outer terms cancel!

Multiply.

Example 2: Finding Products in the Form (a – b)2

A. (x – 6)2

(a – b) = a2 – 2ab + b2

(x – 6) = x2 – 2x(6) + (6)2

= x – 12x + 36

Use the rule for (a – b)2.

Identify a and b: a = x and b = 6.

Simplify.

B. (4m – 10)2

Multiply.

Example 2: Finding Products in the Form (a – b)2

C. (2x – 5y )2

D. (7 – r3)2

Check It Out! Example 2

Multiply.

a. (x – 7)2

b. (3b – 2c)2

Check It Out! Example 2c

Multiply.

(a2 – 4)2

(a + b)(a – b) = a2 – b2

A binomial of the form a2 – b2 is called a difference of two squares.

Multiply.

Example 3: Finding Products in the Form (a + b)(a – b)

A. (x + 4)(x – 4)

(a + b)(a – b) = a2 – b2

(x + 4)(x – 4) = x2 – 42

= x2 – 16

Use the rule for (a + b)(a – b).

Identify a and b: a = x and b = 4.

Simplify.

B. (p2 + 8q)(p2 – 8q)

Multiply.

Example 3: Finding Products in the Form (a + b)(a – b)

C. (10 + b)(10 – b)


Multiply.a. (x + 8)(x – 8)

b. (3 + 2y2)(3 – 2y2)


Multiply.

c. (9 + r)(9 – r)

Review 10.1-10.4 Polynomials

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