Pd Multiplying Polynomials

Section A.8 Multiplying Polynomials

Previously in Section 4.1, we learned how to multiply two monomials together (such as $4xy\cdot3x^2$). And we can add and subtract polynomials even when there is more than one term (such as $(4x^2-3x)+(5x^2+x-2)$) using like terms. In this section, we will learn how to multiply polynomials with more than one term.

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Figure A.8.1. Alternative Video Lesson

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Example A.8.2. Revenue.

Avery owns a local organic jam company that currently sells about $1500$ jars a month at a price of $\$13$ per jar. Avery has found that for each time they would raise the price of a jar by $25$ cents, they will sell $50$ fewer jars of jam per month.

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In general, this company’s revenue can be calculated by multiplying the cost per jar by the total number of jars of jam sold. If we let $x$ represent the number of times the price was raised by $25$ cents, then the price will be $13+0.25x\text{.}$

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At the same time, the number of jars the company sells will be the $1500$ that they currently sell each month, minus $50$ times $x\text{.}$ This gives us the expression $1500-50x$ to represent how many jars the company will sell after raising the price $x$ times.

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Combining these expressions, we can write a formula for the revenue model:

\begin{align*} \text{revenue} \amp= \left(\text{price per item}\right)\times\left(\text{number of items sold}\right)\\ R \amp= \left(13+0.25x\right)\left(1500-50x\right) \end{align*}

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To simplify the expression $\left(13+0.25x\right)\left(1500-50x\right)\text{,}$ we’ll need to multiply $13+0.25x$ by $1500-50x\text{.}$ In this section, we learn how to do that.

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Subsection A.8.1 Review of the Distributive Property

Polynomial multiplication relies on the Distributive Property, and may also rely on the properties of exponents. When we multiply a monomial with a binomial, we can distribute the monomial to each term in the binomial. For example,

\begin{align*} \highlight{-4x}(3x^2+5) \amp= \multiplyleft{(-4x)}\left(3x^2\right)+\multiplyleft{(-4x)}(5)\\ \amp=-12x^3-20x \end{align*}

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Example A.8.3.

A rectangle’s length is $4$ meters longer than its width. Assume its width is $w$ meters. Use a simplified polynomial to model the rectangle’s area in terms of $w$ as the only variable.

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Explanation.

Since the rectangle’s length is $4$ meters longer than its width, we can model its length as $w+4$ meters.

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The rectangle’s area would be:

\begin{align*} A\amp=\ell w\\ \amp=(w+4)w\\ \amp=w^2+4w \end{align*}

The rectangle’s area can be modeled by $w^2+4w$ square meters.

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In the second line of work above, we recognize that $(w+4)w$ is the same as $w(w+4)\text{.}$ Whether the $w$ is written before or after the binomial, we are still able to distribute the $w$ to simplify the product.

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Checkpoint A.8.4.

A rectangle’s length is $4$ feet shorter than $3 \text{ times}$ its width. If we use $w$ to represent the rectangle’s width, use a polynomial to represent the rectangle’s area in expanded form.

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$\displaystyle{ \text{area}=}$ square feet

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Explanation.

The rectangle’s width is $w$ feet. Since the rectangle’s length is $4$ feet shorter than $3 \text{ times}$ its width, its length is $3w-4$ feet. A rectangle’s area formula is:

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$\displaystyle{ \text{area}=(\text{length})\cdot(\text{width}) }$

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After substitution, we have:

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$\displaystyle{\begin{aligned} \text{area} \amp = (\text{length})\cdot(\text{width}) \\ \amp = (3w-4)w \\ \amp ={3w^{2}-4w} \end{aligned} }$

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The rectangle’s area is ${3w^{2}-4w}$ square feet.

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The distributive property can be understood visually with a generic rectangle.

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a rectangle with two regions; the region on the left indicates 2x*3x=6x^2; the region on the right indicates that 2x*4=8x; together they add up to 6x^2+8x — Figure A.8.5. A Generic Rectangle Modeling $2x(3x+4)$
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The big rectangle consists of two smaller rectangles. The big rectangle’s area is $2x(3x+4)\text{,}$ and the sum of those two smaller rectangles is $2x\cdot3x+2x\cdot4\text{.}$ Since the sum of the areas of those two smaller rectangles is the same as the bigger rectangle’s area, we have:

\begin{align*} 2x(3x+4) \amp= 2x\cdot3x+2x\cdot4\\ \amp= 6x^2+8x \end{align*}

Generic rectangles can be used to visualize multiplying polynomials.

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Subsection A.8.2 Multiplying Binomials

Multiplying Binomials Using Distribution.

Whether we’re multiplying a monomial with a polynomial or two larger polynomials together, the first step is still based on the Distributive Property. We’ll start with multiplying two binomials and then move on to examples with larger polynomials.

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We know we can distribute the $3$ in $(x+2)3$ to obtain $(x+2)\multiplyright{3}=x\multiplyright{3}+2\multiplyright{3}\text{.}$ We can actually distribute anything across $(x+2)$ if it is multiplied. For example:

\begin{equation*} (x+2)\cat=x\cdot \cat + 2\cdot \cat \end{equation*}

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Keeping this in mind, we can multiply $(x+2)(x+3)$ by distributing the $(x+3)$ across $(x+2)\text{.}$ Think of $(x+3)$ as the cat.

\begin{equation*} (x+2)\highlight{(x+3)} = x\highlight{(x+3)} + 2\highlight{(x+3)} \end{equation*}

To continue, distribure more:

\begin{align*} (x+2)\highlight{(x+3)} \amp= x\highlight{(x+3)} + 2\highlight{(x+3)}\\ \amp= x \cdot \highlight{x} + x \cdot \highlight{3} + 2 \cdot \highlight{x} + 2 \cdot \highlight{3}\\ \amp=x^2+3x+2x+6\\ \amp=x^2+5x+6 \end{align*}

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To multiply a binomial by another binomial, we simply used distribution twice and then simplified the resulting terms. (Multiplying any two polynomials can be done using these same steps. We will return to that idea later in the section.)

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Multiplying Binomials Using FOIL.

Instead of using two applications of the distributive property, people often memorize a shortcut using the acronym FOIL. The letters refer to the pairs of terms from each binomial that end up multiplied together.

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If we take a closer look at the example we just completed, $(x+2)(x+3)\text{,}$ we can highlight how the FOIL process works. FOIL is the acronym for “First, Outer, Inner, Last”.

\begin{align*} (x+2)(x+3)\amp= (\overbrace{{x} \stackrel{}{\cdot} {x}}^{\text{F}}) + (\overbrace{{3} \stackrel{}{\cdot} {x}}^{\text{O}}) + (\overbrace{{2} \stackrel{}{\cdot} {x}}^{\text{I}}) + (\overbrace{{2} \stackrel{}{\cdot} {3}}^{\text{L}})\\ \amp=x^2+3x+2x+6\\ \amp=x^2+5x+6 \end{align*}

F: $x^2$ 🔗: The $x^2$ term comes from the product of first terms from each binomial.
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O: $3x$ 🔗: The $3x$ term comes from the product of the outer terms from each binomial. This was $x$ in the front of the first binomial and $3$ in the back of the second binomial.
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I: $2x$ 🔗: The $2x$ term comes from the product of the inner terms from each binomial. This was $2$ in the back of the first binomial and $x$ in the front of the second binomial.
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L: $6$ 🔗: The constant term $6$ comes from the product of the last terms of each binomial.
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a diagram that shows how to multiply using FOIL; x*x + x*3 + 2*x + 6 — Figure A.8.6. Using FOIL Method to multiply $(x+2)(x+3)$
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Multiplying Binomials Using Generic Rectangles.

We could approach this same example using the generic rectangle method. To use generic rectangles, we treat $x+2$ as the base of a rectangle, broken up into $x$ and $2\text{.}$ Similarly we treat $x+3$ as the height, broken up into $x$ and $3\text{.}$ Their product, $(x+2)(x+3)\text{,}$ represents the large rectangle’s area.

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a two by two rectangle with the terms x and 2 above the columns and the terms x and 3 on the left side of the rows — Figure A.8.7. Setting up Generic Rectangles to Multiply $(x+2)(x+3)$
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The big rectangle consists of four smaller rectangles. We can find each small rectangle’s area in the next diagram with the area formula for a rectangle: $\text{area}=\text{base}\cdot\text{height}\text{.}$

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the previous generic rectangles with the areas computed; x*x=x^2, x*2=2x, 3*x=3x and 3*2=6 — Figure A.8.8. Using Generic Rectangles to Multiply $(x+2)(x+3)$
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To finish finding this product, we add together the areas of the four smaller rectangles:

\begin{align*} (x+2)(x+3)\amp=x^2+3x+2x+6\\ \amp=x^2+5x+6 \end{align*}

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Notice that the areas of the four smaller rectangles are exactly the same as the four terms we obtained using distribution, which are also the same four terms that came from the FOIL method. The FOIL method and generic rectangles approach are just different ways to represent the distributing.

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Example A.8.9.

Multiply $(2x-3y)(4x-5y)$ using distribution.

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Explanation.

We first distribute the second binomial across $(2x-3y)\text{.}$ Then we’ll distribute again, and simplify the terms that are left.

\begin{align*} (2x-3y)\highlight{(4x-5y)}\amp=2x\highlight{(4x-5y)}-3y\highlight{(4x-5y)}\\ \amp=8x^2-10xy-12xy+15y^2\\ \amp=8x^2-22xy+15y^2 \end{align*}

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Example A.8.10.

Multiply $(2x-3y)(4x-5y)$ using FOIL.

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Explanation.

First, Outer, Inner, Last: Either with arrows on paper or mentally in our heads, we’ll pair up the terms from each factor and multiply those pairs together.

\begin{align*} (2x-3y)(4x-5y)\amp= (\overbrace{{\stackrel{}{2x}}\cdot{4x}}^{\large\text{F}})+ (\overbrace{{\stackrel{}{2x}}\cdot{(-5y)}}^{\large\text{O}})+ (\overbrace{{\stackrel{}{-3y}}\cdot{4x}}^{\large\text{I}})+ (\overbrace{{\stackrel{}{-3y}}\cdot{(-5y}}^{\large\text{L}})\\ \amp=8x^2-10xy-12xy+15y^2\\ \amp=8x^2-22xy+15y^2 \end{align*}

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Example A.8.11.

Multiply $(2x-3y)(4x-5y)$ using generic rectangles.

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Explanation.

We begin by drawing four rectangles and marking their bases and heights with terms in the given binomials:

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a two by two rectangle with the terms 2x and -3y above the columns and the terms 4x and -5y on the left of the rows — Figure A.8.12. Setting up Generic Rectangles to Multiply $(2x-3y)(4x-5y)$
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Next, we calculate each inner rectangle’s area by multiplying its base with its height:

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the previous generic rectangles with the areas multiplied; 2x*4x=8x^2, 2x*-5y=-10xy, -3y*4x=-12xy and -3y*-5y=15y^2 — Figure A.8.13. Using Generic Rectangles to Multiply $(2x-3y)(4x-5y)$
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Finally, we add up all the inner rectangle areas to find the product:

\begin{align*} (2x-3y)(4x-5y)\amp=8x^2-10xy-12xy+15y^2\\ \amp=8x^2-22xy+15y^2 \end{align*}

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Example A.8.14.

Multiply and simplify the formula for Avery’s organic jam revenue, $R$ (in dollars), from Example 2. In that example $R=(13+0.25x)(1500-50x)$ and $x$ represents the number of times they raised the price by 25 cents.

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Explanation.

To multiply this, we’ll use FOIL:

\begin{align*} R \amp= \left(13+0.25x\right)\left(1500-50x\right)\\ \amp= \left(13\cdot1500\right) - \left(13 \cdot 50x \right) + \left( 0.25x \cdot 1500 \right) - \left( 0.25x \cdot 50x \right)\\ \amp= 19500 - 650x + 375x - 12.5x^2\\ \amp= -12.5x^2 - 275x + 19500 \end{align*}

Now we have a formula for Avery’s revenue based on how many times they raise the price. If we wanted to, we could study this formula more to find ways to maximize that revenue.

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Example A.8.15.

Tyrone is an artist and he sells each of his paintings for $\$200\text{.}$ Currently, he can sell $100$ paintings per year. So his annual revenue from selling paintings is $\$200\cdot100=\$20000\text{.}$ He plans to raise the price. However, for each $20 price increase, his customers will buy $5$ fewer paintings each year.

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Assume Tyrone would raise the price of his paintings $x$ times, each time by $20. Use an expanded polynomial to represent his new revenue per year.

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Explanation.

Currently, each painting costs $200. After raising the price $x$ times, each time by $20, each painting’s new price would be $200+20x$ dollars.

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Currently, Tyrone sells $100$ paintings per year. After raising the price $x$ times, each time selling $5$ fewer paintings, he would end up selling $100-5x$ paintings per year.

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His annual revenue can be calculated by multiplying each painting’s price by the number of paintings he would sell:

\begin{align*} \text{annual revenue}\amp=(\text{price})(\text{number of sales})\\ \amp=(200+20x)(100-5x)\\ \amp=200(100)+200(-5x)+20x(100)+20x(-5x)\\ \amp=20000-1000x+2000x-100x^2\\ \amp=-100x^2+1000x+20000 \end{align*}

After raising the price $x$ times, each time by $20, Tyrone’s annual revenue from paintings would be $-100x^2+1000x+20000$ dollars.

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Example A.8.16.

What would happen if we needed to multiply two binomials, but also there is a monomial out front? What is the result for $3(x + 2)(2x + 5)\text{?}$

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Explanation.

We recommend focusing first on the two binomials. We can multiply them using FOIL even though they are written last in the product.

\begin{align*} 3(x + 2)(2x + 5) \amp= 3\left(2x^2 + 5x + 4x + 10\right)\\ \amp= 3\left(2x^2 + 9x + 10\right) \end{align*}

Now we can complete the multiplication just by distributing that $3\text{.}$

\begin{align*} 3(x + 2)(2x + 5) \amp= 6x^2 + 27x + 30 \end{align*}

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Subsection A.8.3 Multiplying Polynomials Larger Than Binomials

To multiply polynomials that have more than two terms, we can use repeated distribution and monomial multiplication. Whether we are working with binomials, trinomials, or even larger polynomials, the process is fundamentally the same.

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Example A.8.17.

Multiply $\left( x+5 \right)\left( x^2-4x+6 \right)\text{.}$

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We can approach this product using either distribution or generic rectangles. We can’t use the FOIL method because there will be more than four pieces. However it can still be helpful to draw arrows for the six pairs of products that will occur.

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Figure A.8.18. Multiply Each Term by Each Term
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We begin by distributing over $\left(x^2-4x+6 \right)\text{,}$ then perform a second step of distributing, and then combine like terms.

\begin{align*} \left(x+5\right)\highlight{\left( x^2-4x+6 \right)}\amp= x\highlight{\left( x^2-4x+6 \right)}+5\highlight{\left( x^2-4x+6 \right)}\\ \amp= x\cdot \highlight{x^2} - x\cdot \highlight{4x} +x\cdot \highlight{6}+5\cdot \highlight{x^2} - 5\cdot \highlight{4x} +5\cdot \highlight{6}\\ \amp= x^3 -4x^2 +6x +5x^2 -20x +30\\ \amp= x^3+x^2-14x+30 \end{align*}

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Checkpoint A.8.19.

Multiply the polynomials.

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$\displaystyle{({a-3b})({a^{2}+7ab+9b^{2}}) = }$

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Explanation.

We multiply the polynomials by using the terms from ${a-3b}$ successively.

\begin{equation*} \begin{aligned} \left({a-3b}\right)\left({a^{2}+7ab+9b^{2}}\right)\amp = {aa^{2}+a\cdot 7ab+a\cdot 9b^{2}-3ba^{2}-3b\cdot 7ab-3b\cdot 9b^{2}}\\ \amp = {a^{3}+4a^{2}b-12ab^{2}-27b^{3}} \end{aligned} \end{equation*}

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Reading Questions A.8.4 Reading Questions

1.

Describe three ways you could go about multiplying $(x+3)(2x+5)\text{.}$

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2.

If you multiplied out $(a+b+c)(d+e+f+g)\text{,}$ how many terms would there be? (Try to answer without actually writing them all down.)

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Exercises A.8.5 Exercises

Review and Warmup

Exercise Group.

Use the properties of exponents to simplify the expression.

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1.

${p^{8}p^{7}}$

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2.

${t^{5}t^{2}}$

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3.

${\left(2v\right)^{3}}$

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4.

${\left(4y\right)^{4}}$

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5.

${4b^{8}\cdot 5b^{9}}$

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6.

${9d^{5}\cdot 7d^{9}}$

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Skills Practice

Multiplying Monomials with Binomials.

Multiply the monomial with the binomial, writing the result as a single simplified polynomial.

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7.

${-5x}\left({x-6}\right)$

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8.

${-3x}\left({x+5}\right)$

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9.

${6x}\left({10x-8}\right)$

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10.

${7x}\left({-5x-10}\right)$

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11.

${4x^{2}}\left({x+4}\right)$

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12.

${7x^{2}}\left({x-8}\right)$

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13.

${6t^{2}}\left({6t^{2}-7t}\right)$

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14.

${-3x^{2}}\left({4x^{2}-2x}\right)$

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15.

${-9x^{2}}\left({10x^{2}-7x-8}\right)$

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16.

${6y^{2}}\left({8y^{2}-2y+3}\right)$

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17.

${-5x^{11}y^{13}}\left({-4x^{19}+9y^{15}}\right)$

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18.

${6x^{12}y^{20}}\left({-8x^{6}-9y^{5}}\right)$

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19.

${-7a^{14}b^{9}}\left({3a^{12}b^{13}+9a^{7}b^{18}}\right)$

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20.

${-8a^{16}b^{17}}\left({7a^{17}b^{3}-9a^{18}b^{5}}\right)$

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21.

${9a^{3}}\left({-10a^{7}+8a^{7}b^{9}-9b^{3}}\right)$

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22.

${-10a^{6}}\left({-5a^{10}-8a^{4}b^{9}+10b^{10}}\right)$

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Multiplying Binomials.

Multiply the binomial with the binomial, writing the result as a single simplified polynomial.

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23.

$\left({x+1}\right)\left({x+9}\right)$

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24.

$\left({x+7}\right)\left({x+3}\right)$

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25.

$\left({5y+7}\right)\left({y+3}\right)$

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26.

$\left({2y+1}\right)\left({y+10}\right)$

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27.

$\left({y+7}\right)\left({y-6}\right)$

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28.

$\left({r+4}\right)\left({r-1}\right)$

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29.

$\left({r-1}\right)\left({r-7}\right)$

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30.

$\left({t-4}\right)\left({t-3}\right)$

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31.

$\left({3t+2}\right)\left({4t+7}\right)$

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32.

$\left({6x+3}\right)\left({4x+1}\right)$

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33.

$\left({4x-1}\right)\left({x-4}\right)$

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34.

$\left({3y-6}\right)\left({5y-4}\right)$

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35.

$\left({9y-2}\right)\left({y-6}\right)$

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36.

$\left({6y-8}\right)\left({y-9}\right)$

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37.

$\left({3r-4}\right)\left({r+10}\right)$

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38.

$\left({9r-10}\right)\left({r+7}\right)$

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39.

$\left({4t-6}\right)\left({3t^{2}-5}\right)$

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40.

$\left({2t-1}\right)\left({t^{2}-5}\right)$

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41.

$\left({9x^{3}+4}\right)\left({x^{2}+8}\right)$

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42.

$\left({5x^{3}+8}\right)\left({x^{2}+6}\right)$

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43.

$\left({2y^{2}-9}\right)\left({3y^{2}-6}\right)$

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44.

$\left({5y^{2}-1}\right)\left({3y^{2}-1}\right)$

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45.

$\left({a-6b}\right)\left({a+10b}\right)$

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46.

$\left({a-7b}\right)\left({a-5b}\right)$

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47.

$\left({a+8b}\right)\left({8a-3b}\right)$

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48.

$\left({a-5b}\right)\left({9a-9b}\right)$

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49.

$\left({10a+7b}\right)\left({2a-5b}\right)$

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50.

$\left({2a-2b}\right)\left({7a+5b}\right)$

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51.

$\left({3ab-6}\right)\left({4ab-5}\right)$

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52.

$\left({4ab+10}\right)\left({10ab+5}\right)$

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53.

${4\mathopen{}\left(y-4\right)\mathopen{}\left(y-1\right)}$

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54.

${3\mathopen{}\left(y+5\right)\mathopen{}\left(y-6\right)}$

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55.

${-5\mathopen{}\left(r-7\right)\mathopen{}\left(r+9\right)}$

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56.

${-3\mathopen{}\left(r+2\right)\mathopen{}\left(r+3\right)}$

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57.

${t\mathopen{}\left(t-6\right)\mathopen{}\left(t+10\right)}$

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58.

${t\mathopen{}\left(t+8\right)\mathopen{}\left(t-2\right)}$

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59.

${0}$

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60.

${-3x\mathopen{}\left(x-5\right)\mathopen{}\left(x+1\right)}$

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61.

${4\mathopen{}\left(4x-3\right)\mathopen{}\left(x-3\right)}$

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62.

${-2\mathopen{}\left(2y+5\right)\mathopen{}\left(y-3\right)}$

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Multiplying Larger Polynomials.

Multiply the polynomials together, writing the result as a single simplified polynomial.

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63.

$\left({-3x+3}\right)\left({-2x^{3}-4x^{2}+5x+5}\right)$

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64.

$\left({4x-5}\right)\left({2x^{3}+4x^{2}+2x+4}\right)$

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65.

$\left({-4x-3}\right)\left({x^{2}-2x-4}\right)$

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66.

$\left({-5x+4}\right)\left({x^{2}+2x-5}\right)$

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67.

$\left({x^{2}+5x+2}\right)\left({x^{2}-2x+5}\right)$

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68.

$\left({x^{2}-2x-4}\right)\left({x^{2}+2x-5}\right)$

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69.

$\left({a-3b}\right)\left({a^{2}-10ab-3b^{2}}\right)$

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70.

$\left({a+4b}\right)\left({a^{2}+5ab+3b^{2}}\right)$

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71.

$\left({a+b-5}\right)\left({a+b+5}\right)$

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72.

$\left({a+b-6}\right)\left({a+b+6}\right)$

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Applications

73.

A rectangle’s length is $6$ feet shorter than $4 \text{ times}$ its width. If we use $w$ to represent the rectangle’s width, use a polynomial to represent the rectangle’s area in expanded form.

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74.

A rectangle’s length is $7$ feet shorter than $3 \text{ times}$ its width. If we use $w$ to represent the rectangle’s width, use a polynomial to represent the rectangle’s area in expanded form.

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75.

A triangle’s height is $8$ feet longer than $6 \text{ times}$ its base. If we use $b$ to represent the triangle’s base, use a polynomial to represent the triangle’s area in expanded form. A triangle’s area can be calculated by $A=\frac{1}{2}bh\text{,}$ where $b$ stands for base, and $h$ stands for height.

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76.

A triangle’s height is $10$ feet longer than $4 \text{ times}$ its base. If we use $b$ to represent the triangle’s base, use a polynomial to represent the triangle’s area in expanded form. A triangle’s area can be calculated by $A=\frac{1}{2}bh\text{,}$ where $b$ stands for base, and $h$ stands for height.

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77.

A trapezoid’s top base is $1$ feet longer than its height, and its bottom base is $3$ feet longer than its height. If we use $h$ to represent the trapezoid’s height, use a polynomial to represent the trapezoid’s area in expanded form. A trapezoid’s area can be calculated by $A=\frac{1}{2}(a+b)h\text{,}$ where $a$ stands for the top base, $b$ stands for the bottom base, and $h$ stands for height.

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78.

A trapezoid’s top base is $2$ feet longer than its height, and its bottom base is $10$ feet longer than its height. If we use $h$ to represent the trapezoid’s height, use a polynomial to represent the trapezoid’s area in expanded form. A trapezoid’s area can be calculated by $A=\frac{1}{2}(a+b)h\text{,}$ where $a$ stands for the top base, $b$ stands for the bottom base, and $h$ stands for height.

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79.

A rectangle’s base can be modeled by ${x+4}$ meters, and its height can be modeled by ${x-3}$ meters. Use a polynomial to represent the rectangle’s area in expanded form.

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80.

A rectangle’s base can be modeled by ${x-5}$ meters, and its height can be modeled by ${x+7}$ meters. Use a polynomial to represent the rectangle’s area in expanded form.

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81.

An artist sells his paintings at ${\$15.00}$ per piece. Currently, he can sell $150$ paintings per year. So his annual income from paintings is $15\cdot150=2250$ dollars. He plans to raise the price. However, for each ${\$2.00}$ of price increase per painting, his customers would buy $9$ fewer paintings annually.

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Assume the artist would raise the price of his painting $x$ times, each time by ${\$2.00}\text{.}$ Use an expanded polynomial to represent his new income per year.

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82.

An artist sells his paintings at ${\$16.00}$ per piece. Currently, he can sell $130$ paintings per year. So his annual income from paintings is $16\cdot130=2080$ dollars. He plans to raise the price. However, for each ${\$3.00}$ of price increase per painting, his customers would buy $7$ fewer paintings annually.

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Assume the artist would raise the price of his painting $x$ times, each time by ${\$3.00}\text{.}$ Use an expanded polynomial to represent his new income per year.

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Challenge

83.

Fill in the blanks with algebraic expressions that make the equation true. You may not use $0$ or $1$ in any of the blank spaces. An example is $\mathord{?} + \mathord{?} = 8x \text{,}$ where one possible answer is $3x + 5x = 8x \text{.}$ There are infinitely many correct answers to this problem. Be creative. After finding a correct answer, see if you can come up with a different answer that is also correct.

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$+$ $= {-15xy}$
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$+$ $= {-13x^{30}y^{5}}$
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$\cdot$ $\cdot$ $\cdot$ $\cdot$ $= {5x^{60}y^{60}}$
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