f(x)=(2x+5)(x+3)(3x-10)(2x-8) has zeros at x=-3,x=-(5)/(2),x=(10)/(3), and x=4. What is the sign of f on the interval -(5)/(2) < x < 4 ? 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.

Factor Analysis: To determine the sign of f(x) on the interval, we need to look at the sign of each factor in the given interval.Sign Determination: For the factor (2x+5), when x is between -(5)/(2) and 4, this factor is positive because 2x+5 is greater than zero for all x in this interval.Factor (2x+5): For the factor (x+3), when x is between -(5)/(2) and 4, this factor is also positive because x+3 is greater than zero for all x in this interval.Factor (x+3): For the factor (3x-10), when x is between -(5)/(2) and 4, this factor is negative because 3x-10 is less than zero when x is less than (10)/(3), which is part of our interval.Factor (3x-10): For the factor (2x-8), when x is between -(5)/(2) and 4, this factor is negative because 2x-8 is less than zero when x is less than 4, which is part of our interval.Factor (2x-8): Since we have two negative factors and two positive factors, the negatives will cancel each other out, and the overall sign of f(x) on the interval will be positive.

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f(x)=(2x+5)(x+3)(3x-10)(2x-8) has zeros at
x=-3,x=-(5)/(2),x=(10)/(3), and
x=4.
What is the sign of
f on the interval
-(5)/(2) < x < 4 ?
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.

$f(x)=(2 x+5)(x+3)(3 x-10)(2 x-8)$ has zeros at $x=-3, x=-\frac{5}{2}, x=\frac{10}{3}$ , and $x=4$ . $\newline$ What is the sign of $f$ on the interval \( -\frac{5}{2}

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Full solution

f(x)=(2 x+5)(x+3)(3 x-10)(2 x-8)

has zeros at

x=-3, x=-\frac{5}{2}, x=\frac{10}{3}

, and

x=4

\newline

What is the sign of

f

on the interval

-\frac{5}{2}<x<4

\newline

Choose

1

answer:

\newline

(A)

f

is always positive on the interval.

\newline

(B)

f

is always negative on the interval.

\newline

(C)

f

is sometimes positive and sometimes negative on the interval.

Factor Analysis: To determine the sign of $f(x)$ on the interval, we need to look at the sign of each factor in the given interval.
Sign Determination: For the factor $(2x+5)$ , when $x$ is between $-\frac{5}{2}$ and $4$ , this factor is positive because $2x+5$ is greater than zero for all $x$ in this interval.
Factor $(2x+5)$ : For the factor $(x+3)$ , when $x$ is between $-\frac{5}{2}$ and $4$ , this factor is also positive because $x+3$ is greater than zero for all $x$ in this interval.
Factor $(x+3)$ : For the factor $(3x-10)$ , when $x$ is between $-\frac{5}{2}$ and $4$ , this factor is negative because $3x-10$ is less than zero when $x$ is less than $\frac{10}{3}$ , which is part of our interval.
Factor $(3x-10)$ : For the factor $(2x-8)$ , when $x$ is between $-\frac{5}{2}$ and $4$ , this factor is negative because $2x-8$ is less than zero when $x$ is less than $4$ , which is part of our interval.
Factor $(2x-8)$ : Since we have two negative factors and two positive factors, the negatives will cancel each other out, and the overall sign of $f(x)$ on the interval will be positive.