Question

Find a polynomial function with the given real zeros whose graph contains the given point. Zeros: -2,0,2 Degree 3 Point: (-4,16)

Short answer

\( f(x) = -\frac{1}{3}(x^3 - 4x) \)

Step-by-step solution

Identify the given zeros

The polynomial has three given real zeros: -2, 0, and 2.

Write the polynomial in factored form

Given the zeros -2, 0, and 2, the polynomial can be written as: \( f(x) = a(x + 2)x(x - 2) \)

Simplify the polynomial

Expand the polynomial by multiplying the factors: \( f(x) = a(x + 2)x(x - 2) = a(x^2 - 4)x = a(x^3 - 4x) \)

Determine the value of the constant 'a'

Use the given point (-4, 16) to find 'a'. Substitute -4 for x and 16 for f(x): \[ 16 = a(-4^3 - 4(-4)) = a(-64 + 16) = a(-48) \] Solving for 'a': \[ a = \frac{16}{-48} = -\frac{1}{3} \]

Write the polynomial function

Substitute the value of 'a' back into the polynomial: \( f(x) = -\frac{1}{3}(x^3 - 4x) \)

Key concepts

Real Zeros

In polynomial functions, 'real zeros' are the values of x for which the polynomial equals zero. In other words, they are the x-intercepts of the polynomial's graph. For example, if you have a polynomial with zeros at -2, 0, and 2, it means the polynomial will touch the x-axis at these points. Understanding real zeros is crucial for graphing polynomial functions, as it helps you know where the graph will intersect the x-axis. Real zeros also aid in writing the polynomial in its factored form.

Factored Form

A polynomial in 'factored form' is written as a product of its factors. This form is particularly useful when you know the zeros of the polynomial. For instance, if the zeros are -2, 0, and 2, the factored form of the polynomial would be: \( f(x) = a(x + 2)x(x - 2) \). Here, each factor corresponds to a zero of the polynomial. The 'a' represents a constant that will later be determined. The factored form makes it easier to expand the polynomial or solve it for specific values of x.

Polynomial Expansion

Expanding a polynomial involves multiplying the factors to convert it from factored form to standard form. Starting with the factored form: \( f(x) = a(x + 2)x(x - 2) \), you first multiply the factors: \( (x + 2)x(x - 2) = x(x^2 - 4) = x^3 - 4x \). Now you have \( f(x) = a(x^3 - 4x) \). Expansion helps in simplifying the polynomial and enables us to substitute values more easily when determining constants.

Determining Constants

The 'constant' in a polynomial, often denoted as 'a', can be determined using a given point. For our exercise, the point (-4, 16) is used, meaning when x = -4, f(x) = 16. Substitute these values into the expanded polynomial: \( f(x) = a(x^3 - 4x) \), so \( 16 = a(-4^3 - 4(-4)) = a(-64 + 16) = a(-48) \). Solve for 'a': \( a = \frac{16}{-48} = -\frac{1}{3} \). Finally, substitute the constant back into the polynomial: \( f(x) = -\frac{1}{3}(x^3 - 4x) \). This step ensures that your polynomial fits the given conditions and point.