Multiplying 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) = A² – B²

Example:

(x + 4)(x – 4) = x²– 16
(5y – 7)(5y + 7) = 25y² - 49

Videos

Multiplying polynomials -
Professor Edward Burger explains multiplying polynomials

This video shows 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

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