Integrated Math 3 / Unit 1: Polynomial Functions
1.3 Factoring and Solving Polynomial Equations
Notes
To find zeros, set $f(x) = 0$ and factor. Then use the zero product property: if $AB = 0$, then $A = 0$ or $B = 0$.
- Factor out the greatest common factor first: $x^3 - 4x = x(x - 2)(x + 2)$.
- Grouping: $x^3 + 2x^2 - 9x - 18 = x^2(x + 2) - 9(x + 2) = (x + 2)(x - 3)(x + 3)$.
- Sum and difference of cubes: $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$ and $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$.
- Quadratic form: $x^4 - 13x^2 + 36 = 0$ is $(x^2 - 4)(x^2 - 9) = 0$, so $x = \pm 2, \pm 3$.
Practice
Solve $x^3 - 4x = 0$.
Factor out $x$: $x(x^2 - 4) = x(x - 2)(x + 2) = 0$.
Factor $x^3 + 8$.
Sum of cubes with $a = x$ and $b = 2$: $(x + 2)(x^2 - 2x + 4)$.
Solve $x^4 - 13x^2 + 36 = 0$.
Let $u = x^2$: $u^2 - 13u + 36 = (u - 4)(u - 9) = 0$. So $x^2 = 4$ or $x^2 = 9$.
Ready for more? Take the Unit 1 test.