Answer :
Final answer:
To identify which given option is not a factor of the function [tex]f(x)=x^3 + 7x^2 -14x -48[/tex], use the factor theorem. Substitute the zero of each potential factor into the function. The one that doesn't result in zero is not a factor.
Explanation:
To find which of the given options is not a factor of the cubic function [tex]f(x)=x^(^3)+7x^(^2^)-14x-48[/tex], we need to use the factor theorem. The factor theorem states that a polynomial f(x) has a factor (x-k) if and only if f(k) = 0.
So, to check if any given option is a factor, we should substitute its zero into the function and see if we get zero as a result. For example, if we were given (x - 3) as a factor, we would substitute 3 into the function. If f(3) = 0, then (x - 3) is a factor.
Any option that doesn't result in zero when substituted is not a factor of the function.
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To determine which of the given expressions is not a factor of the polynomial f(x), we can check if any of them make f(x) equal to zero when substituted into the polynomial. Therefore, the expression (x-2) is not a factor of f(x).
If plugging in a value for x results in f(x) being equal to zero, then that expression is a factor of f(x). Otherwise, it is not a factor.
Let's substitute the values from the given options and check:
For (x-2):
f(2) = (2^3) + 7(2^2) - 14(2) - 48 = 8 + 28 - 28 - 48 = -40
Since f(2) is not equal to zero, (x-2) is not a factor of f(x).
For (x+8):
f(-8) = (-8^3) + 7(-8^2) - 14(-8) - 48 = -512 + 448 + 112 - 48 = 0
Since f(-8) is equal to zero, (x+8) is a factor of f(x).
For (x-3):
f(3) = (3^3) + 7(3^2) - 14(3) - 48 = 27 + 63 - 42 - 48 = 0
Since f(3) is equal to zero, (x-3) is a factor of f(x).
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