Question 1187663: f(x)=x^3+x^2-17x+15
Found 2 solutions by greenestamps, MathLover1:
Answer by greenestamps(13203)   (Show Source):
You can put this solution on YOUR website!
Do you have a question?
Re-post....
The other tutor assumed that what we were supposed to do with this is factor the cubic polynomial.
So she supplied the answer -- which is of absolutely no use to you the student, since her response says NOTHING AT ALL about HOW TO FACTOR the polynomial.
A general cubic polynomial is not going to factor "nicely"; its roots are not likely to be rational.
Assuming they are rational, the typical approach to finding the factorization would be to use the rational roots theorem to find a root, reduce the polynomial to quadratic using synthetic division, and solve the quadratic by factoring.
That path is easy in this example, because it is easy to see that x=1 is a root (f(1)=0), which means (x-1) is a factor.
I'll let you follow that path to the final factorization.
Here is another method for doing the factorization that can be used if we know the roots are integers.
Call the roots a, b, and c.
Vieta's theorem tells us that the sum of the roots is the opposite of the quadratic coefficient divided by the leading coefficient: -1/1 = -1.
It also tells us the product of the roots is the opposite of the constant term divided by the leading coefficient: -15/1 = -15.
So the sum of the roots is -1 and the product is -15. A bit of logical reasoning tells us that there is one negative root and two positive roots. And 15 can be viewed as 1*3*5. And playing with those numbers a bit shows us the roots are 1, 3, and -5: 1+3+(-5)=-1, and (1)(3)(-5)=-15.
And since the roots are 1, 3, and -5, the factorization of the polynomial is...
ANSWER: x^3+x^2-17x+15 = (x-5)(x+1)(x+3)
Answer by MathLover1(20850)  (Show Source):
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