Question 1187663
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Do you have a question?<br>
Re-post....<br>
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The other tutor assumed that what we were supposed to do with this is factor the cubic polynomial.<br>
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.<br>
A general cubic polynomial is not going to factor "nicely"; its roots are not likely to be rational.<br>
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.<br>
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.<br>
I'll let you follow that path to the final factorization.<br>
Here is another method for doing the factorization that can be used if we know the roots are integers.<br>
Call the roots a, b, and c.<br>
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.<br>
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.<br>
And since the roots are 1, 3, and -5, the factorization of the polynomial is...<br>
ANSWER: x^3+x^2-17x+15 = (x-5)(x+1)(x+3)<br>