In these lessons, we will learn how to multiply polynomials.

Related Topics:More Lessons on Algebra

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When finding the product of a monomial and a polynomial, we multiply the monomial by each term of the polynomial. Be careful with the sign (+ or –) of each term.

**Example:**

Evaluate

a) 5(*x* + *y*)

b) – 2*x*(*y* + 3)

c) 5*x*(*x*^{2} – 3)

d) –2*x*^{3}(*x*^{2} – 3*x* + 4)

**Solution:**

a) 5(*x* + *y*) = 5*x* + 5*y
*b) – 2

The multiplication of polynomials having more than one term requires the repeated use of the distributive property.

**Example **

Multiply (2*x* + 5)(*x* +1)

**Solution:**

(2*x* + 5)(*x* + 1)

= *x*(2*x* + 5) + 1(2*x* + 5)

= 2*x*^{2} + 5*x* + 2*x* + 5

= 2*x*^{2} + 7*x* + 5

**Example:**

Multiply (5*y*^{2}– 2*y* + 3)(3*y* – 4)

**Solution:**

(5*y*^{2} – 2*y* + 3)(3*y* – 4)

= 3*y*(5*y*^{2} – 2*y* + 3) – 4(5*y*^{2} – 2*y* + 3)

= 15*y*^{3} – 6*y*^{2} + 9*y* – 20*y*^{2} + 8*y* – 12

= 15*y*^{3} – 26*y*^{2} + 17*y* – 12

We will now consider two special types of products of binomials.

The first type is a trinomial that comes from squaring a binomial. (It is also called a perfect square)

Its general form is

(

A + B)^{2}=A^{2}+ 2AB+B^{2}

**Example:**

(*x* + 5)^{2} = *x*^{2} + 10*x* + 25

(*y* – 3)^{2} = *y*^{2}– 6*y* + 9

The second type is a binomial that comes from multiplying two binomials. (It is also called the difference of two squares)

Its general form is

(

A + B)(A – B) =A^{2}–B^{2}

**Example:**

(*x* + 4)(*x* – 4) = *x*^{2}– 16

(5*y* – 7)(5*y* + 7) = 25*y*^{2} - 49

Multiplying Monomials and/or Binomials and FOIL

We multiply monomials and binomials using different methods, including the distributive property and FOIL. FOIL is a mnemonic device to remember how to find the product of two binomials: we multiply the First, Outer, Inner, and then Last terms in each binomial. When multiplying monomials and binomials, it is important to remember the rules of multiplying exponents.

Special products: Difference of Two Squares and Perfect Square.

This video explains how to multiply monomials and polynomials.

Multiplying Polynomials - Slightly Harder Examples #1

Multiply a trinomial by a trinomial.

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