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.

Multiplying polynomials

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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