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:

Next, use the distributive property over the second binomial:

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:

Next, distribute over the second polynomial:

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.