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Multiplying Polynomials

Site: Saylor Academy
Course: MA007: Algebra
Book: Multiplying Polynomials
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Date: Tuesday, 15 July 2025, 7:37 AM

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Multiplying Polynomials

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 6.45

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

Solution

 The product of a binomial, b plus 3, and a trinomial, 2 b squared minus 5 b plus 8. Two arrows extend from the trinomial, ter
Distribute. The sum of two products, the product of b and 2 b squared minus 5 b plus 8, and the product of 3 and 2 b squared minus 5 b pl
Multiply. \(2 b^{3}-5 b^{2}+8 b+6 b^{2}-15 b+24\)
Combine like terms. \(2 b^{3}+b^{2}-7 b+24\)
Try It 6.89

Multiply using the Distributive Property: \((y-3)\left(y^{2}-5 y+2\right)\).

Try It 6.90

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

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

Example 6.46

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

Solution

It is easier to put the polynomial with fewer terms on the bottom because we get fewer partial products this way.

Multiply \(\left(2 b^{2}-5 b+8\right)\) by \(3\). \( \begin{array}{r} 2 b^{2}-5 b+8 \\ \times \quad b+3 \\ \hline 6 b^{2}-15 b+24 \end{array} \)
  \(\underline{2 b^{3}-5 b^{2}+8 b}\)
Multiply \((2 b^{2}-5 b+8)\) by \(b\). \(2 b^{3}+b^{2}-7 b+24\)
Add like terms.  
Try It 6.91

Multiply using the Vertical Method: \((y-3)(y^{2}-5 y+2)\).

Try It 6.92

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

We have now seen two methods you can use to multiply a trinomial by a binomial. After you practice each method, you'll probably find you prefer one way over the other. We list both methods are listed here, for easy reference.

Multiplying a Trinomial by a Binomial

To multiply a trinomial by a binomial, use the:

  • Distributive Property
  • Vertical Method

Source: OpenStax, https://openstax.org/books/elementary-algebra/pages/6-3-multiply-polynomials
Creative Commons License This work is licensed under a Creative Commons Attribution 4.0 License.

Exercises

Multiply a Trinomial by a Binomial

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

  1. \((x+5)(x^2+4x+3)\)
  2. \((y+8)(4y^2+y−7)\)

In the following exercises, multiply. Use either method.

  1. \((w7)(w^2−9w+10)\)
  2. \((3q+1)(q^2−4q−5)\)

Answers

  1. \(x^3+9x^2+23x+15\)

  2. \(4y^3+33y^2+y−56\)

  3. \(w^3−16w^2+73w−70\)

  4. \(3q^3−11q^2−19q−5\)