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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.

f(x)=(2x1)(3x+5)(x+1) f(x)=(2 x-1)(3 x+5)(x+1) has zeros at x=53,x=1 x=-\frac{5}{3}, x=-1 , and x=12 x=\frac{1}{2} .\newlineWhat is the sign of f f on the interval 53<x<12 -\frac{5}{3}<x<\frac{1}{2} ?\newlineChoose 11 answer:\newline(A) f f is always positive on the interval.\newline(B) f f is always negative on the interval.\newline(C) f f is sometimes positive and sometimes negative on the interval.

Full solution

Q. f(x)=(2x1)(3x+5)(x+1) f(x)=(2 x-1)(3 x+5)(x+1) has zeros at x=53,x=1 x=-\frac{5}{3}, x=-1 , and x=12 x=\frac{1}{2} .\newlineWhat is the sign of f f on the interval 53<x<12 -\frac{5}{3}<x<\frac{1}{2} ?\newlineChoose 11 answer:\newline(A) f f is always positive on the interval.\newline(B) f f is always negative on the interval.\newline(C) f f is sometimes positive and sometimes negative on the interval.
  1. Determine Sign of f(x)f(x): Since f(x)f(x) is a product of three linear factors, we can determine the sign of f(x)f(x) by looking at the signs of each factor in the given interval.
  2. Factor (2x1)(2x-1): First, let's look at the factor (2x1)(2x-1). For xx values between 53-\frac{5}{3} and 12\frac{1}{2}, this factor is negative because 2x2x is always less than 11 in this interval.
  3. Factor (3x+5)(3x+5): Next, the factor (3x+5)(3x+5) is positive for all xx values in the interval, since 3x3x is always greater than 5-5 when xx is greater than 53-\frac{5}{3}.
  4. Factor (x+1)(x+1): Lastly, the factor (x+1)(x+1) is also positive for all xx values in the interval, because xx is always greater than 1-1.
  5. Final Sign of f(x)f(x): Multiplying a negative by two positives gives a negative, so f(x)f(x) is negative for all xx in the interval 53<x<12-\frac{5}{3} < x < \frac{1}{2}.

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