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(x)=x(4x+9)(x−2)(2x−9)(x+5)f, (, x, ), equals, x, (, 4, x, plus, 9, ), (, x, minus, 2, ), (, 2, x, minus, 9, ), (, x, plus, 5, )has zeros at x=-5x=−5x, equals, minus, 5, x=-\dfrac{9}{4}x=− 4 9 ​ x, equals, minus, start fraction, 9, divided by, 4, end fraction, x=0x=0x, equals, 0, x=2x=2x, equals, 2, and x=\dfrac{9}{2}x= 2 9 ​ x, equals, start fraction, 9, divided by, 2, end fraction. What is the sign of fff on the interval 0

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Answer:

The function:

f(x)=x(4x+9)(x−2)(2x−9)(x+5)

has the following zeros: -5, -9/4, 0, 2 and 9/2

Given that f(x) is a polynomial, it is continuous in all real numbers, then from zero to zero it has a different sign.

interval     f(x) sign

(-∞,  -5)      negative, f(-6) = -15120

(-5, -9/4)    positive, f(-4) = 2856  

(-9/4, 0)     negative, f(-1) = -660  

(0, 2)         positive, f(1) = 546

(2, 9/2)      negative, f(3) = -1512

(9/2, ∞)      positive, f(6) = 26136

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