 # Multiplying Polynomials

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

### Multiplying Polynomials and Monomials

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) – 2x(y + 3)
c) 5x(x2 – 3)
d) –2x3(x2 – 3x + 4)

Solution:

a) 5(x + y) = 5x + 5y
b) – 2x(y + 3) = – 2xy – 6x
c) 5x(x2 – 3) = 5x3– 15x
d) –2x3(x2 – 3x + 4) = –2x5 + 6x4 – 8x3

### Multiplying Polynomials and Polynomials

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

Example

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

Solution:

(2x + 5)(x + 1)
= x(2x + 5) + 1(2x + 5)
= 2x2 + 5x + 2x + 5
= 2x2 + 7x + 5

Example:

Multiply (5y2– 2y + 3)(3y – 4)

Solution:

(5y2 – 2y + 3)(3y – 4)
= 3y(5y2 – 2y + 3) – 4(5y2 – 2y + 3)
= 15y3 – 6y2 + 9y – 20y2 + 8y – 12
= 15y3 – 26y2 + 17y – 12

### Multiplying Polynomials – Special Products

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 = A2 + 2AB + B2

Example:

(x + 5)2 = x2 + 10x + 25
(y – 3)2 = y2– 6y + 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) = A2B2

Example:

(x + 4)(x – 4) = x2– 16
(5y – 7)(5y + 7) = 25y2 - 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. How to multiply a binomial and a trinomial by distributing each of the terms in the binomial through the trinomial? Multiplying polynomials
Special products: Difference of Two Squares and Perfect Square. Multiplying Polynomials
This video explains how to multiply monomials and polynomials. Multiplying Polynomials - Slightly Harder Examples #1 Multiplying Polynomials - Slightly Harder Examples #2
Multiply a trinomial by a trinomial.

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