Question

Math

Posted 6 months ago

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$f(x)=(5 x+1)(4 x-8)(x+6)$ 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.
```

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Answer from Sia

Posted 6 months ago

Solution by Steps

step 2

The factor $(5x+1)$ is positive for $x>-0.2$ 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 $-\frac{1}{5}<x<2$, this factor is negative within the entire interval

step 4

The factor $(x+6)$ is positive for all $x>-6$, 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.

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