# Polynomials

### Multiplication of Polynomials

#### 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: 3y 2(12y 3 -6y 2 + 5y - 1) = ?

= 3y 2(12y 3) + (3y 2)(- 6y 2) + (3y 2)(5y) + (3y 2)(- 1)
= (3)(12)y 2+3 + (3)(- 6)y 2+2 + (3)(5)y 2+1 + (3)(- 1)y 2
= 36y 5 -18y 4 +15y 3 -3y 2

Example 2: -4x 3 y(- 2y 2 + xy - x + 9) = ?

= - 4x 3 y(- 2y 2) + (- 4x 3 y)(xy) + (- 4x 3 y)(- x) + (- 4x 3 y)(9)
= (- 4)(- 2)x 3 y 1+2 + (- 4)x 3+1 y 1+1 + (- 4)(- 1)x 3+1 y + (- 4)(9)x 3 y
= 8x 3 y 3 -4x 4 y 2 +4x 4 y - 36x 3 y

#### 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)
= x 2 y + 2xy 2 + 6x + 12y

Example 2. (3x 2 +7)(4 - x 2) = ?
= (3x 2)(4) + (3x 2)(- x 2) + (7)(4) + (7)(- x 2)
= 12x 2 -3x 4 +28 - 7x 2
= - 3x 4 + (12 - 7)x 2 + 28
= - 3x 4 +5x 2 + 28

Example 3: (y - x)(- 4y - 3x) = ?
= (y)(- 4y) + (y)(- 3x) + (- x)(- 4y) + (- x)(- 3x)
= - 4y 2 -3xy + 4xy + 3x 2
= 3x 2 + (- 3 + 4)xy - 4y 2
= 3x 2 + xy - 4y 2

#### Multiplication of Polynomials

The strategy for multiplying two polynomials in general is similar to multiplying two binomials. First, treat the second polynomial as a single term, and distribute over the first term:

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

Next, distribute over the second polynomial:

 a(d + e + f )+ b(d + e + f )+ c(d + e + f )= ad + ae + af + bd + be + bf + cd + ce + cf

At this point, the number of terms in the answer should be the number in the first polynomial times the number in the second polynomial--every combination of a term of the first polynomial and a term of the second polynomial. Since there are 3 terms in each polynomial in this example there should be 3(3) = 9 terms in our answer so far. If the first polynomial had 4 terms and the second had 5 , there would be 4(5) = 20 terms in the answer so far.

Finally, since the the terms in such a product of polynomials are often highly redundant (many have the same variables and exponents), it is important to combine like terms.

Example 1: (x 2 -2)(3x 2 - 3x + 7) = ?
= x 2(3x 2 -3x + 7) - 2(3x 2 - 3x + 7)
= x 2(3x 2) + x 2(- 3x) + x 2(7) - 2(3x 2) - 2(- 3x) - 2(7) ( 6 terms)
= 3x 4 -3x 3 +7x 2 -6x 2 + 6x - 14
= 3x 4 -3x 3 + (7 - 6)x 2 + 6x - 14
= 3x 4 -3x 3 + x 2 + 6x - 14

Example 2: (x 2 + x + 3)(2x 2 - 3x + 1) = ?
= x 2(2x 2 -3x + 1) + x(2x 2 -3x + 1) + 3(2x 2 - 3x + 1)
= x 2(2x 2) + x 2(- 3x) + x 2(1) + x(2x 2) + x(- 3x) + x(1) + 3(2x 2) + 3(- 3x) + 3(1) ( 9 terms)
= 2x 4 -3x 3 + x 2 +2x 3 -3x 2 + x + 6x 2 - 9x + 3
= 2x 4 + (- 3 + 2)x 3 + (1 - 3 + 6)x 2 + (1 - 9)x + 3
= 2x 4 - x 3 +4x 2 - 8x + 3

Note: To check your answer, pick a value for the variable and evaluate both the original expression and your answer--they should be the same.

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