Question

the cubic polynomial below has zeroes at x=-4 and x=6 only and passes through the point (2,36) as shown. algebraically determine its equation in factored form. show how you arrived at your answer.

Answer 1

`f(x)=(1)/(2)(2x-7)(x+4)(x-6)`

Explanation:

Cubic polynomial in intercept form:

`f(x)=(x-p)(x-q)(x-r)`

where p, q and r are the zeros.

Given:

• Zeros at x = -4 and x = 6
• Passes through the point (2, 36)

Substitute the zeros into the formula:

`f(x)=(x-p)(x+4)(x-6)`

Substitute the point into the equation and solve for p:

`\implies (2-p)(2+4)(2-6)=36`

`\implies (2-p)(6)(-4)=36`

`\implies -24(2-p)=36`

`\implies 2-p=-(3)/(2)`

`\implies p=(7)/(2)`

Therefore:

`f(x)=\left(x-(7)/(2)\right)(x+4)(x-6)`

Factor out ¹/₂ from the first parentheses:

`f(x)=(1)/(2)(2x-7)(x+4)(x-6)`