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Find all of the zeros of the polynomial function.

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2 Answers

You do not need a caluclator for this problem.  Group the first two terms together and the second two terms togerter.  Factor out x2 from the first two terms and -5 from the second two terms:

(x3+4x2)+(-5x-20) = x2(x+4)-5(x+4)

Because x+4 is a factor of both groups, it can also be factored out:

x2(x+4)-5(x+4) = (x2-5)(x+4)

Setting g(x)=0, the three values of x which solve this equation are -4, and ±√5.

The way you solve this equation on a calculator can be done many ways.  Using the Ti-89, one could simply use the solve function (F2:1).  This arugement must have an expression, a defined variable.

ex: solve(x^3+4x^2-5x-20=0,x)

I beleive the zeros can be solved for graphically as well on other graphing calculators.  Enter the equation using the 'Y=' menu; fix the window so that the function is visable on the screen [-5,5]x[-20,20]; open the math menu and select the zero function.  You will have to define a lower bound and an upper bound (a point before the desired zero and a point after the zero).  Keep in mind that the zero function solves for one zero at a time.

Regrouping,

g(x) = x2(x+4) - 5(x+4) = (x+4)(x2 - 5) = 0

x = -4, ±√5

If you want to use a calculator, then

Let Y1 = x2(x+4) - 5(x+4)

Finding zeros gives you the answer in decimal form.