Multiplying Polynomials

1
Multiplying Polynomials

• To multiply a polynomial by a monomial,
  distribute the monomial to each term of the
  polynomial and multiply
• (-2x)(4x2 12x 1)
• (-2x)(4x2) (-2x)(12x) (-2x)(-1) ? -8x3 24x2
  2x
• To multiply two polynomials, apply the
  distributive property for each term of either
  polynomial
• (x 2) (x 6) ? x(x 6) 2(x 6) ? x2 6x
  2x 12 ? x2 4x 12
• Find the product of the polynomial shown below
• (a 2b)(a2 b) ? a(a2 b) 2b(a2 b) ? a3
  ab 2a2b 2b2
• (8x x2)(x3 2x2 6)
• (8x)(x3 2x2 6) (x2)(x3 2x2 6)
• 8x4 16x3 48x x5 2x4 6x2 ? x5 10x4
  16x3 6x2 48x

2
Multiplying Polynomials

• The FOIL (First, Outer, Inner, Last) shortcut can
  be used to multiply two binomials
• (x 2) (x 6) ? x2 6x 2x 12 ? x2 4x
  12
• Use FOIL to find the product of these binomials
• (x 2) (x 6) ? x2 6x 2x 12 ? x2 4x
  12
• (x 2) (x 6) ? x2 6x 2x 12 ? x2 8x
  12
• (x 2) (x 2) ? x2 2x 2x 4 ? x2 4
• (x 2) (x 2) ? x2 2x 2x 4 ? x2 4x 4
• (x 2) (x 2) ? x2 2x 2x 4 ? x2 4x 4
• (x 3) (x 3) ? x2 3x 3x 9 ? x2 6x 9
• (3x 2)(3x 2) ? 9x2 6x 6x 4 ? 9x2 12x
  4

3
Multiplying Polynomials

• Recognizing special patterns can make multiplying
  some polynomials easier
• Difference of two squares
• (a b)(a b) ? a2 ab ab b2 ? a2 b2
• Try these
• (x 3)(x 3) (x 4)(x 4) (3x 5)(3x
  5)
• x2 9 x2 16 9x2 25
• Squaring a binomial gives a perfect square
  trinomial
• (a b)2 ? (a b)(a b) ? a2 2ab b2
• (a b)2 ? (a b)(a b) ? a2 2ab b2
• Try these
• (x 3)2 (x 4)2 (2x 3)2
• x2 6x 9 x2 8x 16 4x2 12x 9

4
Multiplying Polynomials

• Recognizing special patterns can make multiplying
  some polynomials easier
• Cube of a binomial
• (a b)3 ? a3 3a2b 3ab2 b3
• (a b)3 ? a3 3a2b 3ab2 b3
• Sum and difference of cubes
• a3 b3 ? (a b)(a2 ab b2)
• a3 b3 ? (a b)(a2 ab b2)