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Polynomials

Multiplication of Polynomials

Problems

Multiplication of Polynomials, page 2

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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

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