a polynomial function is defined as f(x)=ax^3+bx^2+cx+d, where a,b,c, and d are integers. If f(-2)=0 which statement is true?
A. 2 is a zero of f(x)
B. (X-2) must be a factor of f(x)
C. (X+2) must be a factor of f(x)
D. When f(x) is divided by (x-2), the remainder is 0.
the Remainder Theorem says C
we know nothing about f(2) or what happens when f(x) is divided by (x-2)
Well, let's put our thinking caps on and solve this riddle!
If we know that f(-2) = 0, that means when we substitute -2 into the equation, the whole thing equals zero. So let's do some math:
f(-2) = a(-2)^3 + b(-2)^2 + c(-2) + d
Simplifying that, we get:
f(-2) = -8a + 4b - 2c + d
Now, since f(-2) is equal to 0, we can say:
0 = -8a + 4b - 2c + d
And here comes the tricky part. We need to figure out which statement is true based on this equation.
Statement A: 2 is a zero of f(x)
Since f(-2) = 0, this statement is NOT true. We're looking for a zero, not 2.
Statement B: (x - 2) must be a factor of f(x)
This statement implies that when we divide f(x) by (x - 2), we should get zero as the remainder. But we haven't done any division yet, so we can't determine if this statement is true or not.
Statement C: (x + 2) must be a factor of f(x)
Since f(-2) = 0, this statement is true! If we substitute -2 into (x + 2), we get zero.
Statement D: When f(x) is divided by (x - 2), the remainder is 0
Again, we haven't performed any division, so we can't determine if this statement is true or not.
So, the correct answer is C. (x + 2) must be a factor of f(x). Phew! That wasn't as easy as pie, but we got there in the end.
To find the answer, we will determine what value of x satisfies f(x) = 0 when f(x) = ax^3 + bx^2 + cx + d and f(-2) = 0.
If f(-2) = 0, this means that when we substitute x = -2 into the function, the result is zero.
Substituting x = -2 into f(x), we get:
f(-2) = a(-2)^3 + b(-2)^2 + c(-2) + d.
Simplifying further:
f(-2) = -8a + 4b - 2c + d.
Since f(-2) = 0, we have:
0 = -8a + 4b - 2c + d.
Now, let's look at the options given:
A. 2 is a zero of f(x): This is not necessarily true since f(-2) = 0, not f(2).
B. (x - 2) must be a factor of f(x): This is not necessarily true since f(-2) = 0, not f(2).
C. (x + 2) must be a factor of f(x): This is true because when we substitute x = -2, f(-2) = 0, which implies that (x + 2) is a factor of f(x).
D. When f(x) is divided by (x - 2), the remainder is 0: This is not necessarily true since f(-2) = 0, not f(2), and (x - 2) is not a factor of f(x).
Therefore, the correct statement is:
C. (X+2) must be a factor of f(x).
To identify which statement is true, we need to evaluate the given polynomial function with the given value and analyze the resulting equation.
The given polynomial function is f(x) = ax^3 + bx^2 + cx + d, and it is provided that f(-2) = 0.
To determine the value of f(-2), we substitute -2 for x in the equation:
f(-2) = a(-2)^3 + b(-2)^2 + c(-2) + d
Simplifying this equation, we get:
f(-2) = -8a + 4b - 2c + d
Since f(-2) = 0, we can set the equation equal to zero:
-8a + 4b - 2c + d = 0
Now, let's analyze the given statements:
A. 2 is a zero of f(x):
This statement is not necessarily true since the value provided is -2, not 2.
B. (x - 2) must be a factor of f(x):
This statement is not necessarily true since we don't have any information related to the factor (x - 2) based on the given value.
C. (x + 2) must be a factor of f(x):
This statement is true since we know that f(-2) = 0. Substituting -2 for x in the equation, we have f(-2) = a(-2)^3 + b(-2)^2 + c(-2) + d = 0. Therefore, (x + 2) is a factor of f(x).
D. When f(x) is divided by (x - 2), the remainder is 0:
This statement is not necessarily true since there is no information provided about the polynomial being divided by (x - 2).
Therefore, the correct statement is C. (X + 2) must be a factor of f(x).