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# 7.2: Multiplication of Polynomials

Difficulty Level: Advanced Created by: CK-12
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Practice Multiplying Polynomials
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Jack was asked to frame a picture. He was told that the width of the frame was to be 5 inches longer than the glass width and the height of the frame was to be 7 inches longer than the glass height. Jack measures the glass and finds the height to width ratio is 4:3. Write the expression to determine the area of the picture frame.

### Guidance

To multiply polynomials you will need to use the distributive property. Recall that the distributive property says that if you start with an expression like $3(5x+2)$ , you can simplify it by multiplying both terms inside the parentheses by 3 to get a final answer of $15x+6$ .

When multiplying polynomials, you will need to use the distributive property more than once for each problem.

#### Example A

Find the product: $(x+6)(x+5)$

Solution: To answer this question you will use the distributive property. The distributive property would tell you to multiply $x$ in the first set of parentheses by everything inside the second set of parentheses , then multiply 6 in the first set of parentheses by everything in the second set of parentheses . Here is what that looks like:

#### Example B

Find the product: $(2x+5)(x-3)$

Solution: Again, use the distributive property. The distributive property tells you to multiply $2x$ in the first set of parentheses by everything inside the second set of parentheses , then multiply 5 in the first set of parentheses by everything in the second set of parentheses . Here is what that looks like:

#### Example C

Find the product: $(4x+3)(2x^2+3x-5)$

Soltuion: Even though at first this question may seem different, you can still use the distributive property to find the product. The distributive property tells you to multiply $4x$ in the first set of parentheses by everything inside the second set of parentheses, then multiply 3 in the first set of parentheses by everything in the second set of parentheses. Here is what that looks like:

#### Concept Problem Revisited

Jack was asked to frame a picture. He was told that the width of the frame was to be 5 inches longer than the glass width and the height of the frame was to be 7 inches longer than the glass height. Jack measures the glass and finds the height to width ratio is 4:3. Write the expression to determine the area of the picture frame.

What is known?

• The width is 5 inches longer than the glass
• The height is 7 inches longer than the glass
• The glass has a height to width ratio of 4:3

The equations:

• The height of the picture frame is $4x + 7$
• The width of the picture frame is $3x + 5$

The formula:

$\text{Area} &= w \times h \\\text{Area} &= (3x + 5) (4x + 7) \\\text{Area} &= 12x^2 + 21x + 20x + 35 \\\text{Area} &= 12x^2 + 41x + 35$

### Vocabulary

Distributive Property
The distributive property states that the product of a number and a sum is equal to the sum of the individual products of the number and the addends. For example, in the expression: ${\color{red}\frac{2}{3}} ({\color{blue}x + 5})$ , the distributive property states that the product of a number $\left({\color{red}\frac{2}{3}}\right)$ and a sum $({\color{blue}x+5})$ is equal to the sum of the individual products of the number $\left({\color{red}\frac{2}{3}}\right)$ and the addends $({\color{blue}x}$ and ${\color{blue}5})$ .
Like Terms
Like terms refers to terms in which the degrees match and the variables match. For example $3x$ and $4x$ are like terms. Like terms are also known as similar terms .

### Guided Practice

1. Find the product: $(x+3)(x-6)$

2. Find the product: $(2x+5)(3x^2-2x-7)$

3. An average football field has the dimensions of 160 ft by 360 ft. If the expressions to find these dimensions were $(3x+7)$ and $(7x+3),$ what value of $x$ would give the dimensions of the football field?

1. $(x+3)(x-6)$

2. $(2x + 5)(3x^2 - 2x - 7)$

3. $\text{Area} = l \times w$

$\text{Area} &= 360 \times 160 \\(7x+3) &= 360 \\7x &= 360 - 3 \\7x &= 357 \\x &= 51 \\ \\(3x +7) &= 160 \\3x &= 160 - 7 \\3x &= 153 \\x &= 51$

The value of $x$ that satisfies these expressions is 51.

### Practice

Use the distributive property to find the product of each of the following polynomials:

1. $(x+4)(x+6)$
2. $(x+3)(x+5)$
3. $(x+7)(x-8)$
4. $(x-9)(x-5)$
5. $(x-4)(x-7)$
6. $(x+3)(x^2+x+5)$
7. $(x+7)(x^2-3x+6)$
8. $(2x+5)(x^2-8x+3)$
9. $(2x-3)(3x^2+7x+6)$
10. $(5x-4)(4x^2-8x+5)$
11. $9a^2(6a^3+3a+7)$
12. $-4s^2(3s^3+7s^2+11)$
13. $(x+5)(5x^3+2x^2+3x+9)$
14. $(t-3)(6t^3+11t^2+22)$
15. $(2g-5)(3g^3+9g^2+7g+12)$

Apr 30, 2013

Dec 23, 2014