Help with finding x-int of a polynomial!?

Question

f(x)=x^3+x^2-13x-28

How do you find all the zeros of that function? Both imaginary and real. 10 points to the person with the fastest+correct answer cuz this problem is killing me.

Answer 1

f(x) = x^3 + x^2 - 13x - 28

Your first step is to test ALL factors of the constant term, -28. When I say all factors, that includes "plus or minus". The values you want to test for zero are: 1, -1, 2, -2, 4, -4, 7, -7, 14, -14, 28, -28.

Note that you don't have to test every single one; you just have to work your way up and as soon as you find one, you can use long division. Let's demonstrate.

f(1) = 1 + 1 - 13 - 26 = [nonzero], so we reject this possibility.
f(-1) = -1 + 1 + 13 - 28 = [nonzero]; reject.
f(2) = 8 + 4 - 13(2) - 28 = 12 - 26 - 28 = [nonzero]; reject.
f(-2) = -8 + 4 + 26 - 28 = -4 - 2 = [nonzero]; reject.
f(4) = 64 + 16 - 13(4) - 28 = 80 - 52 - 28 = 0

Since 4 is a zero, it follows that (x - 4) is a factor.

At this point, we do synthetic long division;
(x - 4) into (x^3 + x^2 - 13x - 28)

Without showing the details (since long division is difficult to show on here), you should get the answer x^2 + 5x + 7.

This means f(x) factors into (x - 4) (x^2 + 5x + 7)

So if (x - 4) (x^2 + 5x + 7) = 0, then
x - 4 = 0 {i.e. x = 4}
x^2 + 5x + 7 = 0 {i.e. use the quadratic formula}

Using the quadratic formula on x^2 + 5x + 7 = 0, we get

x = [-5 +/- sqrt (25 - 4(7))]/2
x = [-5 +/- sqrt (-3)]/2
x = [-5 +/- sqrt(3)i]/2
x = -5/2 +/- (sqrt(3)/2)i

So your zeros would be:

x = { 4, -5/2 + [sqrt(3)/2]i, -5/2 - [sqrt(3)/2]i }

Answer 2

Use the rational root theorem. The possible rational roots are 1, 2, 4, 7, 14, and 28 and their negatives. By inspection, 4 works. Use synthetic division to factor.

(x - 4)(x^2 + 5x + 7)

Use the quadratic formula for the quadratic factor to get the imaginary roots.
-5/2 +/- i*sqrt(3)/2

The roots are x = 4, -5/2 +/- i*sqrt(3)/2.

Note that the imaginary roots are not x-intercepts of the curve, since you would only graph for real values.

Answer 3

ok the way i was taught which is the easiest is to actually differentiate.as you are dealign with algebra and calculus you should know how to differentiate and integrate. so differentiating the function once will give you the x-ints.

3x^2+2x-13

then using a graphics calculator you can put the equation into equations mode /poly/3 degrees

and you will get the answers

Answer 4

First, test possible factors of 28. x = 4 is a solution.

Therefore,
f(x) = (x-4)(x^2+5x+7) ( I used long division.)
x = (-5±i√24)/2, 4 (three zeros) (by quadratic formula)

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Puggy,

I am done already. :)