6.2.3 Multiplying a Polynomial by a Polynomial

Multiplying a Polynomial by a Polynomial

We have multiplied monomials by monomials, monomials by polynomials, and binomials by binomials. Now we’re ready to multiply a polynomial by a polynomial. Remember, FOIL will not work in this case, but we can use other representations of the Distributive Property such as Vertical Alignment or distributing individual terms or polynomials.

Example 1

Multiply $(b+3)(2b^2-5b+8)$ using the Distributive Property by distributing either a binomial or an individual term.

Step 1 - Distribute.

$(2b^2-5b+8)$ .

$b(2b^2-5b+8)+3(2b^2-5b+8)$

Step 2 - Multiply.

$2b^3-5b^2+8b+6b^2-15b+24$

Step 3 - Combine like terms.

$2b^3+b^2-7b+24$

Example 2

Multiply $(b+3)(2b^2-5b+8)$ using vertical alignment.

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

Step 1 - Multiply.

$(2b^2-5b+8)$ by $3$.

Step 2 - Multiply.

$(2b^2-5b+8)$ by $b$.

Step 3 - Add like terms.

$$\begin{gathered} \underset{―}{\underset{―}{2b^2-5b+8 \\ \times b+3} \\ 6b^2-15b+24 \\ 2b^3-5b^2+8b} \\ 2b^3+b^2-7b+24 \end{gathered}$$