# Polynomials

### Contents

page 1 of 2

Page 1

Page 2

#### Multiplication of a Polynomial by a Monomial

To multiply a polynomial by a monomial, use the distributive property: multiply each term of the polynomial by the monomial. This involves multiplying coefficients and adding exponents of the appropriate variables.

Example 1: 3y2(12y3 -6y2 + 5y - 1) =?

= 3y2(12y3) + (3y2)(- 6y2) + (3y2)(5y) + (3y2)(- 1)
= (3)(12)y2+3 + (3)(- 6)y2+2 + (3)(5)y2+1 + (3)(- 1)y2
= 36y5 -18y4 +15y3 -3y2

Example 2: -4x3y(- 2y2 + xy - x + 9) =?

= - 4x3y(- 2y2) + (- 4x3y)(xy) + (- 4x3y)(- x) + (- 4x3y)(9)
= (- 4)(- 2)x3y1+2 + (- 4)x3+1y1+1 + (- 4)(- 1)x3+1y + (- 4)(9)x3y
= 8x3y3 -4x4y2 +4x4y - 36x3y

#### Multiplication of Binomials

To multiply a binomial by a binomial--(a + b)(c + d ), where a, b, c, and d are terms--use the distributive property twice. First, treat the second binomial as a single term and distribute over the first binomial:

 (a + b)(c + d )= a(c + d )+ b(c + d )

Next, use the distributive property over the second binomial:

 a(c + d )+ b(c + d )= ac + ad + bc + bd

At this point, there should be 4 terms in the answer -- every combination of a term of the first binomial and a term of the second binomial. Simplify the answer by combining like terms.

We can use the word FOIL to remember how to multiply two binomials (a + b)(c + d ):

• Multiply their First terms. (ac)
• Multiply their Outside terms. (ad )
• Multiply their Iinside terms. (bc)
• Multiply their Last terms. (bd )
• Finally, add the results together: ac + ad + bc + bd. Combine like terms.
Remember to include negative signs as part of their respective terms in the multiplication.

Example 1.(xy + 6)(x + 2y) =?
= (xy)(x) + (xy)(2y) + (6)(x) + (6)(2y)
= x2y + 2xy2 + 6x + 12y

Page 1

Page 2