How to Factor & Expand Polynomials

Expanding

• 1

  Write the factors in parentheses side-by-side. If one polynomial has more terms than the other, write the shorter one first.

  (x + 3)(2x ^2 - x + 7)

• 2

  Multiply the first term of the first polynomial by each term in the second polynomial.

  (x + )(2x ^2 - x + 7) = 2x^3 - x^2 +7x

• 3

  Multiply the next term of the first polynomial through the second polynomial. Repeat this step for each additional term in the first polynomial, if necessary.

  ( + 3)(2x ^2 - x + 7) = 6x^2 - 3x +21

• 4

  Combine the solutions and then group like terms together.

  2x^3 - x^2 +7x + 6x^2 - 3x + 21
  2x^3 - x^2 +6x^2 + 7x - 3x + 21

• 5

  Simplify the solution by combining the like functions.

  2x^3 -x^2 +6x^2 + 7x -3x + 21
  (x + 3)(2x ^2 - x + 7) = 2x^3 + 5x^2 + 4x + 21

Factoring

• 6

  Write the polynomial with terms in rank order and then write two sets of parentheses after the equal sign.

  5x - 8 + 3x^2 = 4
  5x - 8 + 3x^2 - 4 = 0
  3x^2 + 5x -12 = ( )( )

• 7

  Factor the first term and put the resulting values in the left side of the parentheses.

  3x^2 = 3x * x
  3x^2 + 5x -12 = (3x )(x )

• 8

  Factor the last term and place the factors in the right-hand side of the parentheses. If more than one set of factors exist, choose one at random.

  -12 = 4 * -3 or 3 * -4
  3x^2 + 5x -12 = (3x + 4)(x - 3)

• 9

  Expand the factor to see if they match the original polynomial.

  3x^2 + 5x -12 = (3x + 4)(x - 3)
  3x^2 + 5x -12 does not equal 3x^2 - 5x - 12

• 10

  Try the next set of factors for the last term if the first set did not work. Continue until you find the correct set.

  3x^2 + 5x -12 = (3x - 4)(x + 3)
  3x^2 + 5x -12 = 3x^2 + 5x -12