The polynomial function shown in the graph crosses the x-axis at `x=-5, x=2 and x=4`. If the equation for this polynomial is written in the form `y=ax^(3)+bx^(2)+cx+d`, with a=1, which of the following could be the equation?

Options

A. `y=x^(3)-x^(2)-38x-40`
B. `y=x^(3)+x^(2)-22x-40`
C. `y=x^(3)-x^(2)-22x+40`
D. `y=x^(3)+5x^(2)+8x+40`

Answer

C

Explanation

Difficult:Medium
Category: Passport to Advanced Math/ Functions
Strategic Advice: You can use the x-intercepts shown in a graph to write the factors of the corresponding polynomial-here, there are three x-intercepts, so you should have three factors.
Getting to the Answer: You are given that a=1, which tells you that each factor is of the form (x-a). To write equation in the form requsted. FOIL the first two factors, then multiply the result by the third factor. The function crosses the x-axis at x=-5, which means one factor is `(x-(-5))`, or `(x_5)`. The function also crosses the x-axis at x=2 and x=4, which translate to factors of `(x-2) and (x-4)`. Start by FOILing `(x+5) and (x-2)`. Then multiply the result by the remaining factors, `(x-4)`:
`(x+5)(x-2)=x^(2)-2x+5x-10=x^(2)+3x-10`
`(x^(2)+3x-10)(x-4)=x^(3)-4x^(2)+3x^(2)-13x-10x+40=x^(3)-x^(2)-22x+40`
This matches (C).