The Factor Theorem and Solving Polynomial Equations

Here's the key rule you need to remember: a polynomial f(x) has a factor $x-a$ if and only if f(a) = 0. This means when you substitute a value into the polynomial and get zero, that value gives you a factor.

To find your first factor, use trial and error with the factors of the constant term. For example, if your constant is 5, try substituting ±1, ±5 into the polynomial until you find one that equals zero.

Once you've found one factor using the factor theorem, you can use polynomial long division to find the remaining factors. This breaks down your cubic equation into simpler parts that are easier to solve.

Quick Tip: Always check the factors of the constant term first - they're your most likely candidates for roots!

Let's look at the example: solving x³ - 4x² + 5 = 0. First, try the factors of 5 (which are ±1, ±5). When you substitute x = -1, you get 0, so $x + 1$ is a factor. Using long division, you can then find that the polynomial equals $x + 1$$x² - 5x + 5$ = 0.

For the quadratic factor x² - 5x + 5 = 0, use the quadratic formula to find the remaining roots: x = 3.618 and x = 1.382 (to 3 decimal places). So your complete solution is x = -1, 3.618, 1.382.