# 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)\begin{align*}3(5x+2)\end{align*}, you can simplify it by multiplying both terms inside the parentheses by 3 to get a final answer of 15x+6\begin{align*}15x+6\end{align*}.

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)\begin{align*}(x+6)(x+5)\end{align*}

Solution: To answer this question you will use the distributive property. The distributive property would tell you to multiply x\begin{align*}x\end{align*} 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)(x3)\begin{align*}(2x+5)(x-3)\end{align*}

Solution: Again, use the distributive property. The distributive property tells you to multiply 2x\begin{align*}2x\end{align*} 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)(2x2+3x5)\begin{align*}(4x+3)(2x^2+3x-5)\end{align*}

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\begin{align*}4x\end{align*} 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\begin{align*}4x + 7\end{align*}
• The width of the picture frame is 3x+5\begin{align*}3x + 5\end{align*}

The formula:

AreaAreaAreaArea=w×h=(3x+5)(4x+7)=12x2+21x+20x+35=12x2+41x+35\begin{align*}\text{Area} &= w \times h \\ \text{Area} &= (3x + 5) (4x + 7) \\ \text{Area} &= 12x^2 + 21x + 20x + 35 \\ \text{Area} &= 12x^2 + 41x + 35\end{align*}

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

### Guided Practice

1. Find the product: \begin{align*}(x+3)(x-6)\end{align*}

2. Find the product: \begin{align*}(2x+5)(3x^2-2x-7)\end{align*}

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

1. \begin{align*}(x+3)(x-6)\end{align*}

2. \begin{align*}(2x + 5)(3x^2 - 2x - 7)\end{align*}

3. \begin{align*}\text{Area} = l \times w\end{align*}

\begin{align*}\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 \end{align*}

The value of \begin{align*}x\end{align*} that satisfies these expressions is 51.

### Practice

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

1. \begin{align*}(x+4)(x+6)\end{align*}
2. \begin{align*}(x+3)(x+5)\end{align*}
3. \begin{align*}(x+7)(x-8)\end{align*}
4. \begin{align*}(x-9)(x-5)\end{align*}
5. \begin{align*}(x-4)(x-7)\end{align*}
6. \begin{align*}(x+3)(x^2+x+5)\end{align*}
7. \begin{align*}(x+7)(x^2-3x+6)\end{align*}
8. \begin{align*}(2x+5)(x^2-8x+3)\end{align*}
9. \begin{align*}(2x-3)(3x^2+7x+6)\end{align*}
10. \begin{align*}(5x-4)(4x^2-8x+5)\end{align*}
11. \begin{align*}9a^2(6a^3+3a+7)\end{align*}
12. \begin{align*}-4s^2(3s^3+7s^2+11)\end{align*}
13. \begin{align*}(x+5)(5x^3+2x^2+3x+9)\end{align*}
14. \begin{align*}(t-3)(6t^3+11t^2+22)\end{align*}
15. \begin{align*}(2g-5)(3g^3+9g^2+7g+12)\end{align*}

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### Vocabulary Language: English

TermDefinition
distributive property The distributive property states that the product of an expression and a sum is equal to the sum of the products of the expression and each term in the sum. For example, $a(b + c) = ab + ac$.

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