Question

Construct a polynomial function with the stated properties. Reduce all fractions to lowest terms.Second-degree, with zeros of - 5 and 3, and goes to -infinity as x -> -infinity

Answer 1

ANSWER

f(x) = -x² - 2x + 15

EXPLANATION

If x1 and x2 are the zeros of a second degree polynomial, the polynomial can be written as,

$P(x)=(x-x_1)(x-x_2)$

In this problem we need to construct a 2nd-degree polynomial function with zeros -5 and 3,

$f(x)=(x+5)(x-3)$

The other property is that f(x) must go to -∞ as x approaches -∞. Therefore, the leading coefficient must be negative, so that the branches of the parabola point downwards,

$f(x)=-(x+5)(x-3)$

Then we can do the multiplication of the factors to obtain the polynomial in standard form,

$f(x)=-(x^2+5x-3x-3\cdot5)$$f(x)=-(x^2+2x-15)$

Distribute the minus sign by changing each coefficient sign,

$f(x)=-x^2-2x+15$

This is a second-degree polynomial function with zeros of -5 and 3 that goes to -∞ as x approaches -∞.

Note: there are other polynomial functions that meet this requirement, this is the simplest one.