Question 814185
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)<br>
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(x) = 2(x-2)(x^2+(-4)x+(-3)x+12)}}}
{{{f(x) = 2(x-2)(x^2+(-7)x+12)}}}
Multiplying the first two factors:
{{{f(x) = (2x-4)(x^2+(-7)x+12)}}}
One more multiplication:
{{{f(x) = 2x*x^2+2x*(-7)x+2x*12+(-4)*x^2+(-4)*(-7)x+(-4)*12)}}}
Simplifying...
{{{f(x) = 2x^3+(-14)x^2+24x+(-4)*x^2+28x+(-48)}}}
{{{f(x) = 2x^3+(-18)x^2+52x+(-48)}}} or {{{f(x) = 2x^3-18x^2+52x-48}}}