Which of the following binomials is a factor of x^3+4x^2+x-6
x-2
x+2
x-4
x+4
But that still didn’t give me the answer
if x-2 is a factor, then x = +2 has to make the expression equal to zero
if x+2 is a factor, then x = -2 has to make the expression equal to zero
try it ...
then use the same concept for the other cases.
To determine which of the following binomials is a factor of x^3 + 4x^2 + x - 6, we can use the factor theorem. According to the factor theorem, if a polynomial function f(x) has a factor of (x - a), then f(a) = 0.
Let's test each of the binomials by plugging them into the polynomial function:
For x - 2:
f(2) = (2)^3 + 4(2)^2 + 2 - 6
= 8 + 16 + 2 - 6
= 20
Since f(2) ≠ 0, x - 2 is not a factor of the given polynomial.
For x + 2:
f(-2) = (-2)^3 + 4(-2)^2 + (-2) - 6
= -8 + 16 - 2 - 6
= 0
Since f(-2) = 0, x + 2 is a factor of the given polynomial.
For x - 4:
f(4) = (4)^3 + 4(4)^2 + 4 - 6
= 64 + 64 + 4 - 6
= 126
Since f(4) ≠ 0, x - 4 is not a factor of the given polynomial.
For x + 4:
f(-4) = (-4)^3 + 4(-4)^2 + (-4) - 6
= -64 + 64 - 4 - 6
= -10
Since f(-4) ≠ 0, x + 4 is not a factor of the given polynomial.
Therefore, the only binomial that is a factor of x^3 + 4x^2 + x - 6 is x + 2.
To determine which of the given binomials is a factor of x^3 + 4x^2 + x - 6, we can use the factor theorem. According to the factor theorem, if a polynomial P(x) is divided by a binomial (x - a), and the remainder is zero, then (x - a) is a factor of P(x).
Let's evaluate each of the given binomials by substituting them into the polynomial and checking if the remainder is zero:
1. Evaluating x - 2:
Substitute x = 2 into the polynomial: (2)^3 + 4(2)^2 + 2 - 6 = 8 + 16 + 2 - 6 = 20.
The remainder is not zero, so x - 2 is not a factor.
2. Evaluating x + 2:
Substitute x = -2 into the polynomial: (-2)^3 + 4(-2)^2 - 2 - 6 = -8 + 16 - 2 - 6 = 0.
The remainder is zero, so x + 2 is a factor.
3. Evaluating x - 4:
Substitute x = 4 into the polynomial: (4)^3 + 4(4)^2 + 4 - 6 = 64 + 64 + 4 - 6 = 126.
The remainder is not zero, so x - 4 is not a factor.
4. Evaluating x + 4:
Substitute x = -4 into the polynomial: (-4)^3 + 4(-4)^2 - 4 - 6 = -64 + 64 - 4 - 6 = -10.
The remainder is not zero, so x + 4 is not a factor.
From the evaluation, we find that only x + 2 gives a remainder of zero when divided into the polynomial. Therefore, the correct answer is x + 2.