1.6 Other Types of Equations, 5.1 Polynomial Functions and Models

Polynomial Equation (Solve by factoring with the grouping method)

Example:  Solve the polynomial equation

y^3+y^2=4y+4

Solution:  Solve the polynomial equation by factoring.

The original equation.
y^3+y^2=4y+4
Write the equation so that all the terms are on the same side.

  • Subtract 4y
  • Subtract 4
y^3+y^2=4y+4
y^3+y^2-4y=4y-4y+4
y^3+y^2-4y=4
y^3+y^2-4y-4=4-4
y^3+y^2-4y-4=0
Group two terms pairs of terms and factor the greatest common factor from each group.

  • The greatest common factor for the first group is y^2
  • The greatest common factor for the second group is -4
y^3+y^2-4y-4=0
(y^3+y^2)+(-4y-4)=0
(y^3+y^2)+(-4y-4)=0
y^2(y+1)+-4(y+1)=0
Factor the common binomial from each term.

  • The common binomial is (y+1)
  • The other factor is formed using the coefficients of the parenthesis
y^2(y+1)+-4(y+1)=0
(y+1)(y^2-4)=0
Continue to factor completely by factoring the difference of squares in the second parenthesis
(y+1)(y^2-4)=0
(y+1)(y+2)(y-2)=0
Apply the zero product property by setting each factor equal to zero.
y+1=0 or y+2=0 or y-2=0
Solve each remaining equation.
y+1=0 or y+2=0 or y-2=0
y+1-1=0-1 or y+2-2=0-2 or y-2+2=0+2
y=-1 or y=-2 or y=2

The solutions to the polynomial equation y^3+y^2=4y+4 are y=-1 or y=-2 or y=2.