Question

Construct a polynomial function with the stated properties. Reduce all fractions to lowest terms. Second-degree, with zeros of −7 and 6, and goes to −∞ as x→−∞.

Answer 1

f( x ) = - x² - x + 42

Explanation:

The polynomial function will have to include the zeroes with opposing signs, considering that when you isolate the value x say, you will take that value to the opposite side, changing the signs,

f(x) = (x + 7)(x - 6)

Now as you can see, x extends to negative infinity, such that,

f(x) = - (x + 7)(x - 6) - that negative makes no difference whatsoever on the zeroes of the function. All we want to do now is to expand this, and we receive out simplified solution.

Goal :
`expand\:-\:\left(x\:+\:7\right)\left(x\:-\:6\right)`,

`- xx+x\left(-6\right)+7x+7\left(-6\right)` =
`- xx-6x+7x-7\cdot \:6` =
`-\left(x^2+x-42\right)`,

Expanded Solution :
`-x^2-x+42`,

Polynomial Function : f( x ) =
`-x^2-x+42`

Answer 2

y = -(x - 6)(x + 7).