Question

Write the simplest polynomial function using the three zeros given, where all coefficients are integers and the leading coefficient is positive. Enter your answers below.

zeros at x = -9, x =
fraction whose numerator is5
and whose denominator is2
end of fraction, , x = 2

Answer 1

The simplest polynomial function can be found by using the given zeros.

First, let's consider the zero at x = -9. Since the zero is at x = -9, it means that (x + 9) is a factor of the polynomial.

Next, let's consider the zero at x = 5/2. Since the zero is at x = 5/2, it means that (2x - 5) is a factor of the polynomial.

To find the polynomial function, we multiply these factors together:

(x + 9)(2x - 5)

Multiplying these factors, we get:

2x^2 + 13x - 45

So, the simplest polynomial function using the given zeros is:

f(x) = 2x^2 + 13x - 45

In this polynomial, the leading coefficient is positive (2), and all the coefficients are integers.

Answer 2

To write a polynomial function using the given zeros, we can use the fact that if a number, say "a," is a zero of a polynomial function, then (x-a) is a factor of the polynomial.

Given that the zeros are -9, 5/2, and 2, we can write the factors as follows:

(x - (-9)) = (x + 9)

(x - (5/2))

(x - 2)

To find the simplest polynomial function, we need to multiply these factors together. However, we need to make sure that the leading coefficient is positive and that all coefficients are integers.

One possible polynomial function is:

f(x) = (x + 9)(2x - 5)(x - 2)

This function has the given zeros, and all coefficients are integers. The leading coefficient is positive since the coefficient of the x^3 term is positive (2).

Another possible polynomial function is:

f(x) = -(x + 9)(-2x + 5)(x - 2)

In this case, the leading coefficient is also positive (-1 * -2 = 2).

Both of these polynomial functions meet the given criteria of having the specified zeros, all integer coefficients, and a positive leading coefficient.