If f, left bracket, x, right bracket, equals, x, cubed, minus, 2, x, squared, minus, 40, x, minus, 64f(x)=x
3
−2x
2
−40x−64, which of the following is not a factor of f, left bracket, x, right bracketf(x)?
Answer
Multiple Choice Answers
left bracket, x, plus, 2, right bracket(x+2)
left bracket, x, plus, 8, right bracket(x+8)
left bracket, x, minus, 8, right bracket(x−8)
left bracket, x, plus, 4, right bracket(x+4)
The answer is (x + 4) is not a factor of f(x).
To determine which of the following is not a factor of f(x), we can use the Remainder Theorem. According to the Remainder Theorem, if a polynomial f(x) is divided by a linear factor (x - a), the remainder will be zero if and only if (x - a) is a factor of f(x).
Let's check each of the given options by dividing f(x) by each factor and see if the remainder is zero.
1. (x + 2):
Dividing f(x) by (x + 2), we get:
f(x) = x^3 - 2x^2 - 40x - 64
Using synthetic division:
-2 | 1 -2 -40 -64
----------------------
1 -4 -32 24
The remainder is 24, so (x + 2) is not a factor of f(x).
2. (x + 8):
Dividing f(x) by (x + 8), we get:
f(x) = x^3 - 2x^2 - 40x - 64
Using synthetic division:
-8 | 1 -2 -40 -64
----------------------
1 -10 80 16
The remainder is 16, so (x + 8) is not a factor of f(x).
3. (x - 8):
Dividing f(x) by (x - 8), we get:
f(x) = x^3 - 2x^2 - 40x - 64
Using synthetic division:
8 | 1 -2 -40 -64
----------------------
1 6 16 24
The remainder is 24, so (x - 8) is not a factor of f(x).
4. (x + 4):
Dividing f(x) by (x + 4), we get:
f(x) = x^3 - 2x^2 - 40x - 64
Using synthetic division:
-4 | 1 -2 -40 -64
----------------------
1 -6 16 48
The remainder is 48, so (x + 4) is not a factor of f(x).
Therefore, the option that is not a factor of f(x) is:
left bracket, x, plus, 4, right bracket (x + 4)
To solve this problem, we can use the factor theorem. The factor theorem states that if a polynomial f(x) has a factor (x-a), then f(a) = 0.
In this case, we need to determine which of the given expressions are not factors of f(x). We can substitute the values in the brackets into f(x) and check if the result is zero.
Let's go through each expression:
1. (x+2): Substitute x = -2 into f(x)
f(-2) = (-2)^3 - 2(-2)^2 - 40(-2) - 64
= -8 - 8 + 80 - 64
= 0
Since f(-2) = 0, this means that (x+2) is a factor of f(x) and it is not the answer.
2. (x+8): Substitute x = -8 into f(x)
f(-8) = (-8)^3 - 2(-8)^2 - 40(-8) - 64
= -512 - 1280 + 320 - 64
= -1536
Since f(-8) is not equal to 0, this means that (x+8) is not a factor of f(x).
3. (x-8): Substitute x = 8 into f(x)
f(8) = (8)^3 - 2(8)^2 - 40(8) - 64
= 512 - 1280 - 320 - 64
= -1152
Since f(8) is not equal to 0, this means that (x-8) is not a factor of f(x).
4. (x+4): Substitute x = -4 into f(x)
f(-4) = (-4)^3 - 2(-4)^2 - 40(-4) - 64
= -64 + 32 + 160 - 64
= 64
Since f(-4) is not equal to 0, this means that (x+4) is not a factor of f(x).
Therefore, the answer is (x+8) (choice 2) since it is the only expression that is not a factor of f(x).