Question

Construct a polynomial function with the stated properties. Reduce all fractions to lowest terms.

Answer 1

The equation of the polynomial equation from the properties is P(x) = 8/15(x + 3)(x + 1)(x - 5)

How to determine the polynomial expression from the stated properties?

The given parameters are

Degree = third degree

Zero = -3, -1 and 5

y-intercept = -8

There are no complex numbers in the above zeros

This means that, the given zeros are the only zeros of the polynomial

i.e Zero = -3, -1 and 5

The equation of the polynomial is then calculated as

P(x) = Leading coefficient * (x - zero)^multiplicity

So, we have

P(x) = a * (x - (-3)) * (x - (-1)) * (x - 5)

This gives

P(x) = a * (x + 3) * (x + 1) * (x - 5)

The y-intercept is -8

This means that

P(0) = -8

i.e. when x = 0, P(x) = -8

So, we have the following representation

a * (0 + 3) * (0 + 1) * (0 - 5) = -8

Evaluate the products

So, we have the following representation

a * -15 = -8

Divide both sides by -5

a = 8/15

Substitute a = 8/15 in P(x) = a * (x + 3) * (x + 1) * (x - 5)

P(x) = 8/15 * (x + 3) * (x + 1) * (x - 5)

So, we have

P(x) = 8/15(x + 3)(x + 1)(x - 5)

Hence, the equation is P(x) = 8/15(x + 3)(x + 1)(x - 5)

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