Multiply Polynomials

By the end of this section, you will be able to: multiply a polynomial by a monomial, multiply a binomial by a binomial, and multiply a trinomial by a binomial.

Multiply a Polynomial by a Monomial

We have used the Distributive Property to simplify expressions like $2(x-3)$ . You multiplied both terms in the parentheses, $x$ 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. Multiply: $4(x+3)$ .

\begin{array}{lrcl} \text{Distribute.} & & & 4 \cdot x + 4 \cdot 3 \\[4pt] \text{Simplify.} & & & 4x + 12 \end{array}

Multiply: $5(x + 7)$ .

Example. Multiply: $y(y-2)$ .

\begin{array}{lrcl} \text{Distribute.} & & & y \cdot y - y \cdot 2 \\[4pt] \text{Simplify.} & & & y^2 - 2y \end{array}

Multiply: $x(x - 7)$ .

Example. Multiply: $7x(2x+y)$ .

\begin{array}{lrcl} \text{Distribute.} & & & 7x \cdot 2x + 7x \cdot y \\[4pt] \text{Simplify.} & & & 14x^2 + 7xy \end{array}

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

Example. Multiply: $-2y\left(4y^2+3y-5\right)$ .

\begin{array}{lrcl} \text{Distribute.} & & & -2y \cdot 4y^2 + (-2y) \cdot 3y - (-2y) \cdot 5 \\[4pt] \text{Simplify.} & & & -8y^3 - 6y^2 + 10y \end{array}

Multiply: $-3y(5y^2 + 8y - 7)$ .

Example. Multiply: $2x^3\left(x^2-8x+1\right)$ .

\begin{array}{lrcl} \text{Distribute.} & & & 2x^3 \cdot x^2 + (2x^3) \cdot (-8x) + (2x^3) \cdot 1 \\[4pt] \text{Simplify.} & & & 2x^5 - 16x^4 + 2x^3 \end{array}

Multiply: $4x(3x^2 - 5x + 3)$ .

When the monomial is the second factor, we use the Distributive Property to multiply, too.

Example. Multiply: $(x+3)\,p$ .

\begin{array}{lrcl} \text{Distribute.} & & & x \cdot p + 3 \cdot p \\[4pt] \text{Simplify.} & & & xp + 3p \end{array}

Multiply: $(x + 8)p$ .

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. We will start by using the Distributive Property.

Using the Distributive Property

Look at the earlier example, where we multiplied a binomial by a monomial.

\begin{array}{lrcl} \text{We distributed the } p \text{ to get:} & & & xp + 3p \\[4pt] \text{What if we have } (x+7) \text{ instead of } p? & & & (x+3)(x+7) \\[4pt] \text{Distribute } (x+7). & & & x(x+7) + 3(x+7) \\[4pt] \text{Distribute again.} & & & x^2 + 7x + 3x + 21 \\[4pt] \text{Combine like terms.} & & & x^2 + 10x + 21 \end{array}

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

Example. Multiply: $(y+5)(y+8)$ .

\begin{array}{lrcl} \text{Distribute } (y+8). & & & y(y+8) + 5(y+8) \\[4pt] \text{Distribute again.} & & & y^2 + 8y + 5y + 40 \\[4pt] \text{Combine like terms.} & & & y^2 + 13y + 40 \end{array}

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

Example. Multiply: $(2y+5)(3y+4)$ .

\begin{array}{lrcl} \text{Distribute } (3y+4). & & & 2y(3y+4) + 5(3y+4) \\[4pt] \text{Distribute again.} & & & 6y^2 + 8y + 15y + 20 \\[4pt] \text{Combine like terms.} & & & 6y^2 + 23y + 20 \end{array}

Multiply: $(3b + 5)(4b + 6)$ .

Example. Multiply: $(4y+3)(2y-5)$ .

\begin{array}{lrcl} \text{Distribute.} & & & 4y(2y-5) + 3(2y-5) \\[4pt] \text{Distribute again.} & & & 8y^2 - 20y + 6y - 15 \\[4pt] \text{Combine like terms.} & & & 8y^2 - 14y - 15 \end{array}

Multiply: $(5y + 2)(6y - 3)$ .

Example. Multiply: $(x-2)(x-y)$ .

\begin{array}{lrcl} \text{Distribute.} & & & x(x-y) - 2(x-y) \\[4pt] \text{Distribute again.} & & & x^2 - xy - 2x + 2y \end{array}

There are no like terms to combine.

Multiply: $(a + 7)(a - b)$ .

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, like in the example $(x-2)(x-y)$ , there are no like terms to combine.

Look at that example again. The product $(x-2)(x-y) = x^2 - xy - 2x + 2y$ has four terms, and each came from multiplying a specific pair:

The First term, $x^2$ , is the product of $x$ and $x$ — the first terms in each binomial.
The Outer term, $-xy$ , is the product of $x$ and $-y$ — the two outer terms.
The Inner term, $-2x$ , is the product of $-2$ and $x$ — the two inner terms.
The Last term, $+2y$ , is the product of $-2$ and $-y$ — 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.

Example. Multiply using the FOIL method: $(x+5)(x+9)$ .

\begin{array}{lrcl} \text{Multiply the First terms.} & & & x^2 \\[4pt] \text{Multiply the Outer terms.} & & & x^2 + 9x \\[4pt] \text{Multiply the Inner terms.} & & & x^2 + 9x + 5x \\[4pt] \text{Multiply the Last terms.} & & & x^2 + 9x + 5x + 45 \\[4pt] \text{Combine like terms.} & & & x^2 + 14x + 45 \end{array}

Multiply using the FOIL method: $(x + 6)(x + 8)$ .

Multiply two binomials using the FOIL method.

Multiply the First terms.
Multiply the Outer terms.
Multiply the Inner terms.
Multiply the Last terms.
Combine like terms, when possible.

Remember, FOIL only works for multiplying two binomials.

Example. Multiply: $(y-7)(y+4)$ .

\begin{array}{lrcl} \text{First:} & & & y^2 \\[4pt] \text{Outer:} & & & y^2 + 4y \\[4pt] \text{Inner:} & & & y^2 + 4y - 7y \\[4pt] \text{Last:} & & & y^2 + 4y - 7y - 28 \\[4pt] \text{Combine like terms.} & & & y^2 - 3y - 28 \end{array}

Multiply: $(x - 7)(x + 5)$ .

Example. Multiply: $(4x+3)(2x-5)$ .

\begin{array}{lrcl} \text{First } (4x \cdot 2x): & & & 8x^2 \\[4pt] \text{Outer } (4x \cdot -5): & & & 8x^2 - 20x \\[4pt] \text{Inner } (3 \cdot 2x): & & & 8x^2 - 20x + 6x \\[4pt] \text{Last } (3 \cdot -5): & & & 8x^2 - 20x + 6x - 15 \\[4pt] \text{Combine like terms.} & & & 8x^2 - 14x - 15 \end{array}

Multiply: $(3x + 7)(5x - 2)$ .

The final products in the last examples were trinomials because we could combine the two middle terms. This is not always the case.

Example. Multiply: $(3x-y)(2x-5)$ .

\begin{array}{lrcl} \text{First:} & & & 6x^2 \\[4pt] \text{Outer:} & & & 6x^2 - 15x \\[4pt] \text{Inner:} & & & 6x^2 - 15x - 2xy \\[4pt] \text{Last:} & & & 6x^2 - 15x - 2xy + 5y \end{array}

There are no like terms to combine.

Multiply: $(7x - y)(2x - 5)$ .

Be careful of the exponents in the next example.

Example. Multiply: $\left(n^2+4\right)(n-1)$ .

\begin{array}{lrcl} \text{First:} & & & n^3 \\[4pt] \text{Outer:} & & & n^3 - n^2 \\[4pt] \text{Inner:} & & & n^3 - n^2 + 4n \\[4pt] \text{Last:} & & & n^3 - n^2 + 4n - 4 \end{array}

There are no like terms to combine.

Multiply: $(x^2 + 6)(x - 8)$ .

Example. Multiply: $(3pq+5)(6pq-11)$ .

\begin{array}{lrcl} \text{First:} & & & 18p^2q^2 \\[4pt] \text{Outer:} & & & 18p^2q^2 - 33pq \\[4pt] \text{Inner:} & & & 18p^2q^2 - 33pq + 30pq \\[4pt] \text{Last:} & & & 18p^2q^2 - 33pq + 30pq - 55 \\[4pt] \text{Combine like terms.} & & & 18p^2q^2 - 3pq - 55 \end{array}

Multiply: $(2ab + 5)(4ab - 4)$ .

Using the Vertical Method

The FOIL method is usually the quickest method for multiplying two binomials, but it only works 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:

\begin{array}{r} 23 \\ \times\, 46 \\ \hline 138 \\ 92\phantom{0} \\ \hline 1058 \end{array}

Start by multiplying $23$ by $6$ to get $138$ . Next, multiply $23$ by $4$ , lining up the partial product in the correct columns. Last, add the partial products. Now we’ll apply this same method to multiply two binomials.

Example. Multiply using the Vertical Method: $(3y-1)(2y-6)$ .

It does not matter which binomial goes on the top. Multiply $3y-1$ by $-6$ , then multiply $3y-1$ by $2y$ , lining up like terms in columns. Finally, add the like terms.

\begin{array}{r} 3y - 1 \\ \times\ 2y - 6 \\ \hline -18y + 6 \\ 6y^2 - 2y\phantom{{}+6} \\ \hline 6y^2 - 20y + 6 \end{array}

Notice the partial products are the same as the terms in the FOIL method.

Multiply using the Vertical Method: $(5m - 7)(3m - 6)$ .

We have now used three methods for multiplying binomials. Be sure to practice each method, and try to decide which one you prefer.

Multiplying two binomials. To multiply binomials, use the:

Distributive Property
FOIL Method
Vertical Method

Remember, FOIL only works when multiplying two binomials.

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, FOIL 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. Multiply using the Distributive Property: $(b+3)\left(2b^2-5b+8\right)$ .

\begin{array}{lrcl} \text{Distribute.} & & & b(2b^2-5b+8) + 3(2b^2-5b+8) \\[4pt] \text{Multiply.} & & & 2b^3 - 5b^2 + 8b + 6b^2 - 15b + 24 \\[4pt] \text{Combine like terms.} & & & 2b^3 + b^2 - 7b + 24 \end{array}

Multiply using the Distributive Property: $(y - 3)(y^2 - 5y + 2)$ .

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

Example. Multiply using the Vertical Method: $(b+3)\left(2b^2-5b+8\right)$ .

It is easier to put the polynomial with fewer terms on the bottom because we get fewer partial products this way. Multiply $2b^2-5b+8$ by $3$ , then multiply $2b^2-5b+8$ by $b$ , lining up like terms, and add.

\begin{array}{r} 2b^2 - 5b + 8 \\ \times\phantom{2b^2 - 5}\ b + 3 \\ \hline 6b^2 - 15b + 24 \\ 2b^3 - 5b^2 + 8b\phantom{{}+24} \\ \hline 2b^3 + b^2 - 7b + 24 \end{array}

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

Multiplying a trinomial by a binomial. To multiply a trinomial by a binomial, use the:

Distributive Property
Vertical Method

Key terms

FOIL method — a shortcut for multiplying two binomials by adding the products of the First, Outer, Inner, and Last pairs of terms. Vertical Method — a way to multiply any two polynomials by stacking them and adding partial products, just like multiplying whole numbers. Both the FOIL and Vertical methods are organized applications of the Distributive Property.

This section is adapted from Elementary Algebra 2e, Section 6.3: Multiply Polynomials by Lynn Marecek, MaryAnne Anthony-Smith, and Andrea Honeycutt Mathis, © OpenStax, licensed under CC BY-NC-SA 4.0. Access the original for free at openstax.org. Changes: recast the worked-example step tables as typeset display arrays and the Vertical Method partial-product work as stacked-multiplication arrays; described the FOIL first/outer/inner/last diagrams in prose; omitted the Be Prepared quiz, Self Check checklist, media links, and end-of-section exercises; and converted the practice problems (“Try Its”) into interactive exercises with instant feedback.

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