Given that x=1/3 is a zero of f(x)=3x^3−x^2−27x+9, apply the Factor Theorem and factor f(x) into linear factors.(1 point)
f(x)=(x^2−9)(3x−1)
f(x)=(x+3)(x−3)(3x−1)
f(x)=(x+3)(x−3)(3x+1)
f(x)=(x−3)(3x^2−8x−3)
The correct answer is f(x)=(x−3)(3x^2−8x−3).
To apply the Factor Theorem, we need to find the factors of f(x) that are represented by the zero x=1/3.
The Factor Theorem states that if x=a is a zero of a polynomial f(x), then (x-a) is a factor of f(x).
Given that x=1/3 is a zero of f(x)=3x^3−x^2−27x+9, we can write:
f(x) = (x - 1/3) * [some polynomial]
To find the remaining polynomial, we need to divide f(x) by (x - 1/3) using long division or synthetic division.
Using long division:
3x^2 + 8x - 9
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x - 1/3 | 3x^3 - x^2 - 27x + 9
-(3x^3 - x^2 + 9x)
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- 26x + 9
- (- 26x + 8)
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1
Therefore, the remaining polynomial is 3x^2 + 8x - 9.
Combining the factor (x - 1/3) with the remaining polynomial, we have:
f(x) = (x - 1/3) * (3x^2 + 8x - 9)
So, the correct answer is:
f(x) = (x - 1/3) * (3x^2 + 8x - 9)