how to you solve polynomial equation by factoring
x^3 + 3x^2 - 9x = 27
x^3 + 3x^2 - 9x - 27 = 0
looks like grouping will work
x^2(x+3) - 9(x+3) = 0
(x+3)(x^2 - 9) = 0
(x+3)(x+3)(x-3) = 0
x = ± 3
(x = -3 is a double root)
To solve a polynomial equation by factoring, follow these steps:
Step 1: Set the equation equal to zero by subtracting the constant term from both sides:
x^3 + 3x^2 - 9x - 27 = 0
Step 2: Group the terms to look for common factors. In this case, you can group the first two terms and the last two terms:
(x^3 + 3x^2) - (9x + 27) = 0
Step 3: Factor out the greatest common factor from each group. In the first group, you can factor out an x^2, and in the second group, you can factor out a 9:
x^2(x + 3) - 9(x + 3) = 0
Step 4: Notice that the terms in parentheses, (x + 3), are the same. Factor out this common factor:
(x^2 - 9)(x + 3) = 0
Step 5: The next step is to solve each factor, set them equal to zero and solve separately:
For the first factor, x^2 - 9 = 0, you can use the difference of squares rule:
(x - 3)(x + 3) = 0
This gives two possible solutions: x = 3 and x = -3.
For the second factor, x + 3 = 0:
x = -3
So the solutions to the equation x^3 + 3x^2 - 9x = 27 are x = -3 and x = 3