Question

Find a polynomial function with the given real zeros whose graph contains the given point. Zeros: -4,-1,2 Degree 3 \(y\) -intercept: 16

Short answer

The polynomial function is \(f(x) = -2(x + 4)(x + 1)(x - 2)\).

Step-by-step solution

Write the General Form of the Polynomial

A polynomial of degree 3 with zeros at \(-4\), \(-1\), and \(2\) can be written in the factored form as \(f(x) = a(x + 4)(x + 1)(x - 2)\). Here, \a\ is a constant coefficient.

Apply the Given Point to Find \(a\)

Given that the graph contains the y-intercept \(16\), substituting \(x = 0\) and \y = 16\ into the polynomial equation gives \f(0) = a(0 + 4)(0 + 1)(0 - 2) = 16\. This simplifies to \a \cdot 4 \cdot 1 \cdot (-2) = 16\ or \[-8a = 16\].

Solve for \(a\)

Solving for \(a\) gives: \a = -2\.

Write the Specific Form of the Polynomial

Substitute \(a = -2\) back into the polynomial equation to get: \f(x) = -2(x + 4)(x + 1)(x - 2)\.

Key concepts

real zeros

Real zeros of a polynomial function are the values of \(x\) where the function equals zero. These are also known as the roots or solutions of the equation. For example, if a polynomial has real zeros at \( -4, -1\), and \( 2\), this means that the graph of the function crosses the \(x\)-axis at these points. Finding the real zeros of a polynomial is essential for sketching its graph and understanding its behavior.

To find real zeros:

• Set the polynomial function equal to zero: \(f(x) = 0\).
• Solve the equation for \(x\).
• The solutions will be the real zeros.

Knowing the real zeros is the first step in writing the polynomial in its factored form.

degree of polynomial

The degree of a polynomial is the highest power of the variable \(x\) that appears in the polynomial when it is expressed in its standard form. For example, in the polynomial \(f(x) = -2(x + 4)(x + 1)(x - 2)\), expanding it would show that the highest power of \(x\) is 3, making it a third-degree (or cubic) polynomial. The degree of the polynomial tells us:

• The maximum number of real zeros.
• The behavior of the graph as \(x\) approaches positive or negative infinity.
• The number of turning points, which is one less than the degree.

factored form

The factored form of a polynomial is a way of expressing the polynomial as a product of its factors. Each factor corresponds to a real zero of the polynomial. For example, if a polynomial function has zeros at \( -4, -1\), and \(2\), and is of degree 3, it can be written as:\[f(x) = a(x + 4)(x + 1)(x - 2)\] Here, \(a\) is a constant which can be found using additional information like a given point on the graph.

Factoring is useful because:

• It makes it easier to find the zeros of the polynomial.
• It helps in simplifying the polynomial for graphing and solving purposes.
• It reveals the multiplicity of the zeros (how many times a particular factor appears).

constant coefficient

The constant coefficient \(a\) in the polynomial equation affects the stretch or compression of the graph. It is determined using a known point on the graph. For example, if the polynomial has a y-intercept of \(16\), we substitute \(x = 0\) and \(y = 16\) into the equation \[f(x) = a(x + 4)(x + 1)(x - 2)\] This results in solving for \(a\):
\[16 = a \times 4 \times 1 \times (-2)\]
\[-8a = 16\]
\[a = -2\]
Substituting this back into the polynomial gives the specific equation: \[f(x) = -2(x + 4)(x + 1)(x - 2)\]

The constant coefficient:

• Scales the polynomial's graph.
• Ensures the graph passes through particular points.
• Reflects the polynomial, if negative.