SOLUTION: Find a polynomial satisfying the following conditions: f has degree 3;the roots of f are precisely 2,3,4;the leading coefficient is 2.

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: Find a polynomial satisfying the following conditions: f has degree 3;the roots of f are precisely 2,3,4;the leading coefficient is 2.      Log On


   

Question 814185: Find a polynomial satisfying the following conditions:
f has degree 3;the roots of f are precisely 2,3,4;the leading coefficient is 2.
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
If "r" is a root of a polynomial, then (x - r) is a factor of the polynomial. So with the three given roots and the leading coefficient we can immediately write a factored version of the desired function:
f(x) = 2(x-2)(x-3)(x-4)

All that is left is to multiply this out. Since this is all multiplication we can multiply in any order. I'll start by using FOIL on the last two factors:
f(x) = 2(x-2)(x*x+x*(-4)+(-3)*x+(-3)(-4))
Simplifying...
f%28x%29+=+2%28x-2%29%28x%5E2%2B%28-4%29x%2B%28-3%29x%2B12%29
f%28x%29+=+2%28x-2%29%28x%5E2%2B%28-7%29x%2B12%29
Multiplying the first two factors:
f%28x%29+=+%282x-4%29%28x%5E2%2B%28-7%29x%2B12%29
One more multiplication:

Simplifying...
f%28x%29+=+2x%5E3%2B%28-14%29x%5E2%2B24x%2B%28-4%29%2Ax%5E2%2B28x%2B%28-48%29
f%28x%29+=+2x%5E3%2B%28-18%29x%5E2%2B52x%2B%28-48%29 or f%28x%29+=+2x%5E3-18x%5E2%2B52x-48