Factor and Solve Polynomial Equations
Factor and Solve Polynomial Equations
Goal: Solving polynomial equations by factoring.
Polynomial Identities
In order to solve polynomials we need to be aware of basic identities which allow us to simplify the equation.
The following are the main identities for binomial factors
2.) x^{3} + y^{3} = (x + y)(x^{2} − xy + y^{2})
3.) x^{3} − y^{3} = (x − y)(x^{2} + xy + y^{2})
4.) x^{4} − y^{4} = (x − y)(x + y)(x^{2} + y^{2})
Solve by Factoring
Example
Solve for x in the equation by factoring and find all the complex roots: 2x^{3} + 16 = 0
Solution
Since 2 is a common factor to each term, factor out 2:
2(x^{3} + 8) = 0
Then factor the remaining cubic expression:
2(x + 2)(x^{2} − 2x + 4) = 0
Use the zero-product property:
2 = 0 OR x + 2 = 0 OR x^{2} − 2x + 4 = 0
Solve each equation:
2 = 0 2 ≠ 0, so this one gets discarded | x + 2 = 0 x = −2 | x^{2} − 2x + 4 = 0 x = = |
and we find the solutions