To determine which of the given options is a factor of the polynomial x^3 - 5x^2 - 2x + 24, we can use synthetic division.
The given linear factor x - 3 implies that x = 3 satisfies the polynomial, so we perform synthetic division using 3 as the divisor:
3 | 1 -5 -2 24
-----------------
3 -6 -24 0
The result of the synthetic division is 3x^2 - 6x - 24, with no remainder. This means that (x - 3) is a factor of x^3 - 5x^2 - 2x + 24.
To determine if the other options are factors, we can substitute them into the polynomial and check if the result is zero.
a) Substitute x = 2 into the polynomial:
(2)^3 - 5(2)^2 - 2(2) + 24 = 8 - 20 - 4 + 24 = 8 ≠ 0
b) Substitute x = -4 into the polynomial:
(-4)^3 - 5(-4)^2 - 2(-4) + 24 = -64 - 80 + 8 + 24 = -112 ≠ 0
c) Substitute x = -2 into the polynomial:
(-2)^3 - 5(-2)^2 - 2(-2) + 24 = -8 - 20 + 4 + 24 = 0
Therefore, the factor of x^3 - 5x^2 - 2x + 24 is x + 2.
Which of the following is a factor of x^3−5x^2−2x+24, given that x−3 is one of the linear factors?
x-2
x+4
x^2-8x+22
x+2
1 answer