10.3 Multiply Polynomials

Learning objectives

Learning objectives

• Multiply a polynomial by a monomial

• Multiply a binomial by a binomial

• Multiply a trinomial by a binomial

Multiply a Polynomial by a Monomial

In Distributive Property you learned to use the Distributive Property to simplify expressions such as $2(x-3).$ You multiplied both terms in the parentheses, $x\,\mathrm{and}\,3,$ by $2,$ to get $2x-6.$ With this chapter's new vocabulary, you can say you were multiplying a binomial, $x-3,$ by a monomial, $2.$ Multiplying a binomial by a monomial is nothing new for you!

Example 10.28

Solution

1. Distribute the factor across the sum. Multiply $3$ by each term inside the parentheses to obtain $3x+3(7)$.
2. Simplify each product. The first product is $3x$, and the second is $3\cdot{}7=21$.
3. Combine the resulting terms. The expression becomes $3x+21$, with no like terms to combine further.
4. State the product. The arrows in the figure show $3$ multiplying both terms, and the simplified result is $3x+21$.

Takeaway: Distribute a monomial to every term inside parentheses, and remember that the parentheses do not stop multiplication.

Example 10.29

Solution

1. Distribute the outside monomial. Multiply $x$ by each term in $x-8$ to get $x(x)+x(-8)$.
2. Simplify the first product. The factor $x\cdot{}x$ uses the Product Property of Exponents, giving ${x}^{2}$.
3. Simplify the second product. The multiplication $x(-8)$ produces $-8x$, since a negative coefficient times $x$ is $-8x$.
4. State the product. The figures display $x(x-8)$, the distributive arrows, and the result ${x}^{2}-8x$.

Takeaway: When distributing a monomial over a difference, multiply both the positive and negative terms inside the parentheses.

Example 10.30

Solution

1. Distribute $10x$ across the trinomial. Write the product as $(10x)(4x)+(10x)y$.
2. Simplify the first product. Multiplying coefficients and adding exponents gives $(10\cdot{}4){x}^{1+1}=40{x}^{2}$.
3. Simplify the second product. The product $(10x)y$ is $10xy$, with the variables in the order shown.
4. State the result. The figures end with the expanded polynomial $40{x}^{2}+10xy$, which contains no like terms to combine.

Takeaway: Distribute a monomial across every term of a trinomial, then use exponent rules on each product separately.

Multiplying a monomial by a trinomial works in much the same way.

Example 10.31

Solution

1. Distribute $-2x$ to each term. Write $(-2x)(5{x}^{2})+(-2x)(7x)+(-2x)(-3)$.
2. Simplify the quadratic product. The product is $(-2\cdot{}5){x}^{1+2}=-10{x}^{3}$.
3. Simplify the linear and constant products. The second product is $(-2\cdot{}7){x}^{2}=-14{x}^{2}$, while the third is $(-2)(-3)x=6x$.
4. State the expanded polynomial. The result is $-10{x}^{3}-14{x}^{2}+6x$, and no like terms remain to combine.

Takeaway: A negative coefficient is distributed to every term, so each product inherits the sign before simplification.

Example 10.32

Solution

1. Distribute $4{y}^{3}$ across the trinomial. The product becomes $(4{y}^{3})({y}^{2})+(4{y}^{3})(-8y)+(4{y}^{3})(1)$.
2. Simplify the first two products. The first is $4{y}^{5}$ because $3+2=5$, and the second is $-32{y}^{4}$ because $3+1=4$.
3. Simplify the constant product. Multiplying by $1$ leaves $4{y}^{3}$ unchanged.
4. State the expanded result. The expression is $4{y}^{5}-32{y}^{4}+4{y}^{3}$, with no like terms to combine.

Takeaway: The order of the factors does not change distribution, and each exponent is added only within its own product.

Now we will have the monomial as the second factor.

Example 10.33

Solution

1. Distribute $p$ across the sum. Rewrite $(x+3)p$ as $p(x)+p(3)$, since multiplication distributes over addition in either factor order.
2. Simplify the first product. The multiplication $p\cdot{}x$ gives the term $px$.
3. Simplify the second product. The multiplication $p\cdot{}3$ gives the term $3p$.
4. Check the signs and combine. Both products are positive, and the terms $px$ and $3p$ have different variable parts.
5. State the result. The two partial products form the simplified expression $px+3p$.

Takeaway: The distributive property works no matter which factor is written outside the parentheses.

Multiply a Binomial by a Binomial

Just like there are different ways to represent multiplication of numbers, there are several methods that can be used to multiply a binomial times a binomial.

Using the Distributive Property

We will start by using the Distributive Property. Look again at Example 10.33.

Notice that before combining like terms, we had four terms. We multiplied the two terms of the first binomial by the two terms of the second binomial—four multiplications.

Be careful to distinguish between a sum and a product.

Example 10.34

Solution

1. Distribute the first term. Multiply $x$ by each term of the second binomial to obtain $x(x)+x(8)$.
2. Distribute the second term. Multiply $6$ by each term of the second binomial to obtain $6(x)+6(8)$.
3. Combine the four products. Simplifying gives ${x}^{2}+8x+6x+48$.
4. Combine like terms and state the result. The linear terms add to $14x$, so the product is ${x}^{2}+14x+48$; the figures display the same four partial products.

Takeaway: Multiplying binomials by distribution creates four terms, one for every pairing of the two terms in each binomial.

Now we'll see how to multiply binomials where the variable has a coefficient.

Example 10.35

Solution

1. Distribute the first binomial’s terms. Write the product as $(2x)(3x+4)+9(3x+4)$.
2. Distribute the second term. Expand to $(2x)(3x)+(2x)(4)+9(3x)+9(4)$.
3. Simplify all four products. They are $6{x}^{2}+8x+27x+36$.
4. Combine like terms and state the result. The linear terms give $35x$, so the product is $6{x}^{2}+35x+36$; the figure highlights the common factor used to begin the expansion.

Takeaway: In FOIL expansion, multiply coefficients and variables separately for each of the four products before combining.

In the previous examples, the binomials were sums. When there are differences, we pay special attention to make sure the signs of the product are correct.

Example 10.36

Solution

1. Distribute the first binomial’s terms. Rewrite the product as $4y(6y-5)+3(6y-5)$.
2. Distribute within both parts. This gives $(4y)(6y)-(4y)(5)+3(6y)-3(5)$.
3. Simplify each product. The four results are $24{y}^{2}-20y+18y-15$.
4. Combine like terms and state the result. The linear terms give $-2y$, so the product is $24{y}^{2}-2y-15$; the figure shows the shared factor before expansion.

Takeaway: Keep each subtraction visible while distributing so that a negative factor changes the sign of its entire product.

Up to this point, the product of two binomials has been a trinomial. This is not always the case.

Example 10.37

Solution

1. Distribute the first term. Multiply $x$ by each term of $x-y$ to get $x(x)+x(-y)$.
2. Distribute the second term. Multiply $2$ by each term of $x-y$ to get $2(x)+2(-y)$.
3. Simplify the four products. The terms become ${x}^{2}-xy+2x-2y$.
4. Combine like terms and state the result. No terms are like, so the product remains ${x}^{2}-xy+2x-2y$, matching the final expression in the figures.

Takeaway: The order of terms in FOIL does not change the product, but preserving each sign is essential when combining.

Using the FOIL Method

Remember that when you multiply a binomial by a binomial you get four terms. Sometimes you can combine like terms to get a trinomial, but sometimes there are no like terms to combine. Let's look at the last example again and pay particular attention to how we got the four terms.

Where did the first term, ${x}^{2},$ come from?

It is the product of $x\,\mathrm{and}\,x,$ the first terms in $(x+2)\,\mathrm{and}\,(x-y).$

The next term, $-\mathrm{xy},$ is the product of $x\,\mathrm{and}\,-y,$ the two outer terms.

The third term, $+2x,$ is the product of $2\,\mathrm{and}\,x,$ the two inner terms.

And the last term, $-2y,$ came from multiplying the two last terms.

We abbreviate “First, Outer, Inner, Last” as FOIL. The letters stand for ‘First, Outer, Inner, Last’. The word FOIL is easy to remember and ensures we find all four products. We might say we use the FOIL method to multiply two binomials.

Let's look at $(x+3)(x+7)$ again. Now we will work through an example where we use the FOIL pattern to multiply two binomials.

Example 10.38

Solution

1. Multiply the First terms. The product $x\cdot{}x$ is ${x}^{2}$.
2. Multiply the Outer terms. The product $x\cdot{}9$ is $9x$, so the partial expansion is ${x}^{2}+9x$.
3. Multiply the Inner and Last terms. The Inner product is $6\cdot{}x=6x$, and the Last product is $6\cdot{}9=54$.
4. Combine like terms and state the result. The expansion ${x}^{2}+9x+6x+54$ combines to ${x}^{2}+15x+54$, as shown in the final figures.

Takeaway: FOIL computes First, Outer, Inner, and Last products in order, and like terms are combined only at the end.

We summarize the steps of the FOIL method below. The FOIL method only applies to multiplying binomials, not other polynomials!

Example 10.39

Solution

1. Multiply the First and Outer terms. The First product is $y\cdot{}y={y}^{2}$, and the Outer product is $y\cdot{}6=6y$.
2. Multiply the Inner term. The product $(-8)\cdot{}y$ is $-8y$.
3. Multiply the Last term. The product $(-8)\cdot{}6$ is $-48$, giving the expansion ${y}^{2}+6y-8y-48$.
4. Combine like terms and state the result. The linear terms give $6y-8y=-2y$, so the product is ${y}^{2}-2y-48$, matching the final figure.

Takeaway: Assign the four FOIL labels before multiplying so that negative coefficients produce the correct signs.

Example 10.40

Solution

1. Multiply the First terms. The product $(2a)(3a)$ is $6{a}^{2}$.
2. Multiply the Outer terms. The product $(2a)(-1)$ is $-2a$.
3. Multiply the Inner and Last terms. The Inner product $3(3a)$ is $9a$, and the Last product $3(-1)$ is $-3$.
4. Combine like terms and state the result. The expansion $6{a}^{2}-2a+9a-3$ becomes $6{a}^{2}+7a-3$, which agrees with the final figure.

Takeaway: With coefficients in a FOIL product, multiply coefficients and powers in each term before adding like terms.

Example 10.41

Solution

1. Multiply the First terms. The product $(5x)(2x)$ is $10{x}^{2}$.
2. Multiply the Outer terms. The product $(5x)(-7)$ is $-35x$.
3. Multiply the Inner and Last terms. The Inner product $(-y)(2x)$ is $-2xy$, and the Last product $(-y)(-7)$ is $7y$.
4. State the expanded result. The four unlike terms give $10{x}^{2}-35x-2xy+7y$, which matches the figures and needs no combining.

Takeaway: FOIL gives four products even when the binomials use different variables, so terms with unlike variables remain separate.

Using the Vertical Method

The FOIL method is usually the quickest method for multiplying two binomials, but it works only for binomials. You can use the Distributive Property to find the product of any two polynomials. Another method that works for all polynomials is the Vertical Method. It is very much like the method you use to multiply whole numbers. Look carefully at this example of multiplying two-digit numbers.

You start by multiplying $23$ by $6$ to get $138.$

Then you multiply $23$ by $4,$ lining up the partial product in the correct columns.

Last, you add the partial products.

Now we'll apply this same method to multiply two binomials.

Example 10.42

Solution

1. Set up the vertical multiplication. Place $(5x-1)$ above $(2x-7)$ with like powers aligned in the same columns.
2. Multiply by the $5x$ row. The two partial products are $5x\cdot{}2x=10x^2$ and $5x\cdot(-7)=-35x$.
3. Multiply by the $-1$ row. The two partial products are $-1\cdot{}2x=-2x$ and $(-1)(-7)=7$.
4. Add like terms. The $x^2$ coefficient is $10$, the $x$ coefficient is $-35-2=-37$, and the constant is $7$.
5. State the product. The aligned partial products combine to give $10x^2-37x+7$.

Takeaway: The vertical method works because each partial product corresponds to one term of FOIL, and aligning like powers yields $(5x-1)(2x-7)=10x^2-37x+7$.

We have now used three methods for multiplying binomials. Be sure to practice each method, and try to decide which one you prefer. The three methods are listed here to help you remember them.

Multiply a Trinomial by a Binomial

We have multiplied monomials by monomials, monomials by polynomials, and binomials by binomials. Now we're ready to multiply a trinomial by a binomial. Remember, the FOIL method will not work in this case, but we can use either the Distributive Property or the Vertical Method. We first look at an example using the Distributive Property.

Example 10.43

Solution

1. Distribute $x$ across the trinomial. This gives $x(2{x}^{2})+x(-5x)+x(8)$.
2. Distribute $3$ across the trinomial. This gives $3(2{x}^{2})+3(-5x)+3(8)$.
3. Simplify all six products. They become $2{x}^{3}-5{x}^{2}+8x+6{x}^{2}-15x+24$.
4. Combine like terms and state the result. The $x^2$ terms give ${x}^{2}$ and the $x$ terms give $-7x$, so the product is $2{x}^{3}+{x}^{2}-7x+24$.

Takeaway: Multiplying a binomial by a trinomial gives six partial products, two for each term of the trinomial.

Now let's do this same multiplication using the Vertical Method.

Example 10.44

Solution

1. Set up the vertical multiplication. Place $2{x}^{2}-5x+8$ on top and $x+3$ below it, then multiply by $3$ for the first row.
2. Compute the first partial product. Multiplying each term by $3$ gives $6{x}^{2}-15x+24$.
3. Compute the shifted partial product. Multiplying by $x$ gives $2{x}^{3}-5{x}^{2}+8x$, and shifting one place aligns its powers with those of the first row.
4. Add the partial products and state the result. Combining like powers gives $2{x}^{3}+{x}^{2}-7x+24$, as shown in the final figures.

Takeaway: In vertical polynomial multiplication, align equal powers and shift each successive partial product by one place.

Section 10.3 Exercises

Section 10.3 Exercises

Multiply a Polynomial by a Monomial

In the following exercises, multiply.

Multiply a Binomial by a Binomial

In the following exercises, multiply the following binomials using: ⓐ the Distributive Property ⓑ the FOIL method ⓒ the Vertical method

In the following exercises, multiply the following binomials. Use any method.

Multiply a Trinomial by a Binomial

In the following exercises, multiply using ⓐ the Distributive Property and ⓑ the Vertical Method.

In the following exercises, multiply. Use either method.

Everyday Math

Writing Exercises

Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?

1. Multiply: $6(x+8).$

2. Multiply: $2(y+12).$

3. Multiply: $y(y-9).$

4. Multiply: $p(p-13).$

5. Multiply: $8x(x+3y).$

1. Multiply: $3r(6r+s).$

2. Multiply: $-4y(8{y}^{2}+5y-9).$

3. Multiply: $-6x(9{x}^{2}+x-1).$

4. Multiply: $3{x}^{2}(4{x}^{2}-3x+9).$

5. Multiply: $8{y}^{2}(3{y}^{2}-2y-4).$

1. Multiply: $(x+8)p.$

2. Multiply: $(a+4)p.$

3. Multiply: $(x+8)(x+9).$

4. Multiply: $(a+4)(a+5).$

5. Multiply: $(5x+9)(4x+3).$

1. Multiply: $(10m+9)(8m+7).$

2. Multiply: $(7y+1)(8y-3).$

3. Multiply: $(3x+2)(5x-8).$

4. Multiply: $(x+5)(x-y).$

5. Multiply: $(x+2y)(x-1).$

1. Multiply using the FOIL method: $(x+7)(x+8).$

2. Multiply using the FOIL method: $(y+14)(y+2).$

3. Multiply: $(y-3)(y+8).$

4. Multiply: $(q-4)(q+5).$

5. Multiply: $(4a+9)(5a-2).$

1. Multiply: $(7x+4)(7x-8).$

2. Multiply: $(12x-y)(x-5).$

3. Multiply: $(6a-b)(2a-9).$

4. Multiply using the vertical method: $(4m-9)(3m-7).$

5. Multiply using the vertical method: $(6n-5)(7n-2).$

1. Multiply using the Distributive Property: $(y-1)({y}^{2}-7y+2).$

2. Multiply using the Distributive Property: $(x+2)(3{x}^{2}-4x+5).$

3. Multiply using the Vertical Method: $(y-1)({y}^{2}-7y+2).$

4. Multiply using the Vertical Method: $(x+2)(3{x}^{2}-4x+5).$

5. $4(x+10)$

1. $15(r-24)$

2. $-3(m+11)$

3. $-8(z-5)$

4. $u(u+5)$

5. $n({n}^{2}-3n)$

1. $12x(x-10)$

2. $-9a(3a+5)$

3. $6x(4x+y)$

4. $5p(11p-5q)$

5. $3({v}^{2}+10v+25)$

1. $2n(4{n}^{2}-4n+1)$

2. $-8y({y}^{2}+2y-15)$

3. $5{q}^{3}({q}^{2}-2q+6)$

4. $-4{z}^{2}(3{z}^{2}+12z-1)$

5. $(2y-9)y$

1. $(w-6)\,\cdot\,8$

2. $(x+4)(x+6)$

3. $(n+12)(n-3)$

4. $(y+8)(y+3)$

5. $(a+6)(a+16)$

1. $(u-5)(u-9)$

2. $(z-10)(z-22)$

3. $(x-4)(x+7)$

4. $(v+12)(v-5)$

5. $(6n+5)(n+1)$

1. $(2m-9)(10m+1)$

2. $(4c-1)(4c+1)$

3. $(3u-8)(5u-14)$

4. $(a+b)(2a+3b)$

5. $(5x-y)(x-4)$

1. $(u+4)({u}^{2}+3u+2)$

2. $(a+10)(3{a}^{2}+a-5)$

3. $(y-6)({y}^{2}-10y+9)$

4. $(2x+1)({x}^{2}-5x-6)$

5. Mental math You can use binomial multiplication to multiply numbers without a calculator. Say you need to multiply $13$ times $15.$ Think of $13$ as $10+3$ and $15$ as $10+5.$

1. ⓐ Multiply $(10+3)(10+5)$ by the FOIL method.

2. ⓑ Multiply $13\,\cdot\,15$ without using a calculator.

3. ⓒ Which way is easier for you? Why?

1. Which method do you prefer to use when multiplying two binomials—the Distributive Property, the FOIL method, or the Vertical Method? Why?