7.2: Multiplication of Polynomials
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.
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Khan Academy Multiplying Polynomials
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 \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 \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: \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 \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: \begin{align*}(2x+5)(x-3)\end{align*}
Solution: Again, use the distributive property. The distributive property tells you to multiply \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: \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 \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 \begin{align*}4x + 7\end{align*}
- The width of the picture frame is \begin{align*}3x + 5\end{align*}
The formula:
\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?
Answers:
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:
- \begin{align*}(x+4)(x+6)\end{align*}
- \begin{align*}(x+3)(x+5)\end{align*}
- \begin{align*}(x+7)(x-8)\end{align*}
- \begin{align*}(x-9)(x-5)\end{align*}
- \begin{align*}(x-4)(x-7)\end{align*}
- \begin{align*}(x+3)(x^2+x+5)\end{align*}
- \begin{align*}(x+7)(x^2-3x+6)\end{align*}
- \begin{align*}(2x+5)(x^2-8x+3)\end{align*}
- \begin{align*}(2x-3)(3x^2+7x+6)\end{align*}
- \begin{align*}(5x-4)(4x^2-8x+5)\end{align*}
- \begin{align*}9a^2(6a^3+3a+7)\end{align*}
- \begin{align*}-4s^2(3s^3+7s^2+11)\end{align*}
- \begin{align*}(x+5)(5x^3+2x^2+3x+9)\end{align*}
- \begin{align*}(t-3)(6t^3+11t^2+22)\end{align*}
- \begin{align*}(2g-5)(3g^3+9g^2+7g+12)\end{align*}
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Here you will learn how to multiply polynomials using the distributive property.