f(x)=(2x-1)(3x+5)(x+1) has zeros at x=-(5)/(3),x=-1, and x=(1)/(2). What is the sign of f on the interval -(5)/(3)

Question

f(x)=(2x-1)(3x+5)(x+1) has zeros at  x=-(5)/(3),x=-1, and  x=(1)/(2). What is the sign of  f on the interval  -(5)/(3) < x < (1)/(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.

Bytelearn · Accepted Answer

Determine Sign of f(x): Since f(x) is a product of three linear factors, we can determine the sign of f(x) by looking at the signs of each factor in the given interval.Factor (2x-1): First, let's look at the factor (2x-1). For x values between -(5)/(3) and (1)/(2), this factor is negative because 2x is always less than 1 in this interval.Factor (3x+5): Next, the factor (3x+5) is positive for all x values in the interval, since 3x is always greater than -5 when x is greater than -(5)/(3).Factor (x+1): Lastly, the factor (x+1) is also positive for all x values in the interval, because x is always greater than -1.Final Sign of f(x): Multiplying a negative by two positives gives a negative, so f(x) is negative for all x in the interval -(5)/(3) < x < (1)/(2).

$f(x)=(2 x-1)(3 x+5)(x+1)$ has zeros at $x=-\frac{5}{3}, x=-1$ , and $x=\frac{1}{2}$ . $\newline$ What is the sign of $f$ on the interval \( -\frac{5}{3}

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$f(x)=(2 x-1)(3 x+5)(x+1)$ has zeros at $x=-\frac{5}{3}, x=-1$ , and $x=\frac{1}{2}$ . $\newline$ What is the sign of $f$ on the interval \( -\frac{5}{3}