Question

Determine whether each binomial is a factor of x
3 + 3x
2
− 10x − 24.

8. x + 4

9. x − 3

10. x + 6

11. The volume, in cubic inches, of a rectangular box can be expressed as the
product of its three dimensions and with the function V(x) = x
3 − 16x
2 + 79x −
120. The length of the box is represented by the expression x − 8. Find linear
expressions with integer coefficients for the width and height. Hint: The width is
greater than the height.
12. What does it mean if P(−4) for the polynomial function
P(x) = x
3 + 11x
2 + 34x + 24 = 0?

Answer 1

Given P(x) = x^3 + 3x^2 − 10x − 24.

8. x + 4
P(−4) = (−64) + 48 + 40 − 24 = 0 → yes, x + 4 is a factor.

9. x − 3
P(3) = 27 + 27 − 30 − 24 = 0 → yes, x − 3 is a factor.

10. x + 6
P(−6) = −216 + 108 + 60 − 24 = −72 ≠ 0 → no, x + 6 is not a factor.

(For reference, P(x) = (x + 4)(x − 3)(x + 2).)

11. V(x) = x^3 − 16x^2 + 79x − 120, length = x − 8 so x = 8 is a root. Divide by (x − 8) (synthetic division):
Coefficients 1, −16, 79, −120 with 8 gives quotient x^2 − 8x + 15 = (x − 3)(x − 5).
Width and height (integer-coefficient linear factors): x − 3 and x − 5. Since width > height, width = x − 3, height = x − 5.

12. If P(−4) = 0 for P(x) = x^3 + 11x^2 + 34x + 24, it means x = −4 is a root (zero) of P(x), so (x + 4) is a factor and the graph crosses (or touches) the x-axis at x = −4. (Indeed P(x) = (x + 4)(x + 1)(x + 6).)