Polynomial Question?

Question

The graph of f(x) = ax^4+ bx^2 + cx -24 crosses the x-axis at 1,-2 and 3. Determine the equation of f(x).

 

How do I get an answer of f(x)=-4x^3 + 8x^2 + 20x - 24

Answer 1

Well, if 1, -2 and 3 are roots of your polynomial function, then, working backwards, (x - 1), (x + 2) and (x - 3) must be factors. Your equation must look something like 

(x - 1)(x + 2)(x - 3) = 0 which multiplied out is

(x^2 + x - 2)(x - 3) = 0 which is

x^3 - 2x^2 - 5x + 6 = 0.

 

But YOUR last term was -24.....so multiply both sides by -4 to get....

 

-4x^3 + 8x^2 + 20x - 24 = 0

 

So p(x) = -4x^3 + 8x^2 + 20x - 24

 

Let's check.

p(1) = -4 + 8 + 20 - 24 = 0 therefore this graph goes through (1,0) which is an x intercept of 1

p(-2) = 32 + 32 - 40 - 24 = 0 therefore this graph goes through (-2,0) which is an x intercept of -2

p(3) = -108 + 72 + 60 -24 = 0 therefore this graph goes through (3,0) which is an x intercept of 3.

 

So this answer works.