To solve a polynomial equation, we need to find the values of x that make the polynomial 0. That is, we want to find the roots of the polynomial. Then, if we want to give it a root canal, we'll know where to start.
If the polynomial factors into polynomials of degree 1, we can find the roots by factoring the polynomial. Ah, it feels good to stretch our factoring muscles once again. It's been five minutes; they were starting to cramp.
Solve x2 – 4x – 5 = 0.
We can factor the polynomial as
(x – 5)(x + 1).
In order for this product to equal zero, one of the factors must equal zero. Either
x – 5 = 0 and so x = 5
x + 1 = 0 and so x = -1.
The roots of the polynomial, which are the solutions to the equation, are x = 5 and x = -1.
When a polynomial doesn't factor nicely, it can be hard to find its roots, even if you do extensive research on Ancestry.com. We'll talk about ways to find roots for some other polynomials later, so hold onto your hat.
Solve x3 - 5x2 + 6x = 0.
Solve x2 - 25 = 0.
Solve - x2 - 8x + 9.