Question

If x^3 - 5x - 4x + 20 is going to factor, then one of these combinations is going to “work”, that is, the polynomial will divide evenly and give f(k) = 0 according to the Factor Theorem. So the best thing to do is employ a little trail and error. Let’s start with the smaller numbers, they will be easier to evaluate.

1. Substitute for x: 1, -1, 2, -2, 4, -4 ...20, -20.

Answer 1

Explanation:

x^3 - 5x^2 - 4x + 20 (I'm assuming this is what you meant)

(1)^3 - 5(1)^2 - 4(1) + 20 = 1 - 5 - 4 + 20 = 12

(-1)^3 - 5(-1)^2 - 4(-1) + 20 = -1 - 5 + 4 + 20 = 18

(2)^3 - 5(2)^2 - 4(2) + 20 = 8 - 20 - 8 + 20 = 0

Now I know (x-2) is a factor of the polynomial.

Synthetic division

2 | 1. -5. -4. 20

| 2. -6. -20

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1. -3. -10. 0

x^2 -3x -10

Factor -> (x-5)(x+2)

So the factors of the original polynomial are (x-5), (x+2), and (x-2)

f(x) = 0 when x = -2,2,or 5