8/20/2019 C_Solving a Polynomial Inequality

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Kwadwo Amankwa - 07/04/2015 4:43:30 AM PDTPreCalculus / Amankwa, Kwadwo (Amankwa)

Solving a polynomial inequality

Solve the inequality.

( + 4)( − 4)(2 − 4) ≥ 0

Write your answer as an interval or union of intervals.

We'll use the fact that a polynomial only changes signs at its real zeros.

That is, a polynomial is always positive or always negative between two consecutive real

zeros.

So, we'll find the real zeros of the polynomial ( + 4)( − 4)(2 − 4).

( + 4)( − 4)(2 − 4) = 0

(

+ 4)(

− 4)(

+ 2)(

− 2) = 0

real zeros:    = − 4,    = 4,    = − 2, and  = 2

These four zeros split the real numbers into the following five intervals.

( − ∞ , − 4) , ( − 4, − 2) , ( − 2, 2) , (2, 4) , (4, ∞ )

We'll choose a test value in each interval and evaluate the polynomial at that value.

The sign of the polynomial at the test value tells us the sign for the entire interval.

Interval Test value () Value of (+ 4)(−4)(+2)(− 2)

( − ∞ , − 4) −5 (−5 + 4)(−5 − 4)(−5 + 2)(−5 − 2) >  

( − 4, − 2) −3 (−3 + 4)(−3 − 4)(−3 + 2)(−3 − 2) < 0

( − 2, 2) 0 (0 + 4)(0 − 4)(0 + 2)(0 − 2) >  

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