$f(x)=(5 x+1)(4 x-8)(x+6)$ has zeros at $x=-6, x=-\frac{1}{5}$, and $x=2$.

What is the sign of $f$ on the interval $-\frac{1}{5}

Q: $f(x)=(5 x+1)(4 x-8)(x+6)$ has zeros at $x=-6, x=-\frac{1}{5}$, and $x=2$. What is the sign of $f$ on the interval $-\frac{1}{5}<x<2$ ? Choose 1 answer: (A) $f$ is always positive on the interval. (B) $f$ is always negative on the interval. (c) $f$ is sometimes positive and sometimes negative on the interval.

∻Solution by Steps∻ ‖ step 1 ⋮ To determine the sign of $f(x)$ on the interval $-\frac{1}{5} -0.2$ since $-0.2$ is its zero and the leading coefficient is positive. ‖ ‖ step 3 ⋮ The factor $(4x-8)$ is positive for $x>2$, but since our interval is $-\frac{1}{5} -6$, which includes our entire interval of interest. ‖ ‖ step 5 ⋮ Since two factors are positive and one is negative within the interval, their product $f(x)$ is negative for $-\frac{1}{5}<x<2$. ‖ ∻[question number] Answer∻ ***B*** ∻Key Concept∻ ⚹Sign of a polynomial function on an interval⚹ ∻Explanation∻ ⚹The sign of a polynomial function within an interval can be determined by examining the sign of each of its factors over that interval. If there is an odd number of negative factors, the function is negative; if there is an even number of negative factors, the function is positive.⚹◊What are the locations of the critical points in the interval $-\frac{1}{5}

Question

$f(x)=(5 x+1)(4 x-8)(x+6)$ has zeros at $x=-6, x=-\frac{1}{5}$, and $x=2$.

What is the sign of $f$ on the interval $-\frac{1}{5}<x<2$ ?
Choose 1 answer:
(A) $f$ is always positive on the interval.
(B) $f$ is always negative on the interval.
(c) $f$ is sometimes positive and sometimes negative on the interval.

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ To determine the sign of $f(x)$ on the interval $-\frac{1}{5}-0.2$ since $-0.2$ is its zero and the leading coefficient is positive. ‖ ‖ step 3 ⋮ The factor $(4x-8)$ is positive for $x>2$, but since our interval is $-\frac{1}{5}-6$, which includes our entire interval of interest. ‖ ‖ step 5 ⋮ Since two factors are positive and one is negative within the interval, their product $f(x)$ is negative for $-\frac{1}{5}

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