Question

Find a polynomial function with the given real zeros whose graph contains the given point. Zeros: -4 (multiplicity 1), 0 (multiplicity 3), 2 (multiplicity 1); degree 5; contains the point (-2,64)

Short answer

The polynomial is \[ f(x) = -(x + 4)x^3(x - 2) \]

Step-by-step solution

- Write the form of the polynomial

A polynomial with the given roots and multiplicities would be: \[ f(x) = a(x + 4)(x^3)(x - 2) \] Since the degree is 5, the form of the polynomial is correct. Remember to include the leading coefficient 'a'.

- Substitute the given point into the polynomial

Use the point (-2, 64) to find 'a'. Substitute -2 for x and 64 for f(x) into the polynomial: \[ 64 = a(-2 + 4)(-2)^3(-2 - 2) \]

- Simplify the equation

Simplify the right side of the equation: \[ 64 = a(2)(-8)(-4) \] Continue to simplify: \[ 64 = a(-64) \]

- Solve for 'a'

Solve for 'a' by dividing both sides of the equation by -64: \[ 64 = -64a \]\[ a = -1 \]

- Write the final polynomial

Now that 'a' is -1, substitute it back into the polynomial: \[ f(x) = -1(x + 4)x^3(x - 2) \]Simplify if necessary: \[ f(x) = -(x + 4)x^3(x - 2) \]

Key concepts

Real Zeros

Real zeros of a polynomial are the values of x where the polynomial equals zero. These are the points where the graph of the polynomial intersects the x-axis. For the given polynomial, we have three real zeros: -4, 0, and 2. Each zero provides a factor of the polynomial:

• (x + 4) for zero at x = -4
• x for zero at x = 0
• (x - 2) for zero at x = 2

These zeros tell us a lot about the shape and behavior of the polynomial. When plotted, each real zero corresponds to a place where the polynomial crosses or touches the x-axis.

Multiplicity

Multiplicity refers to how many times a particular real zero occurs in the polynomial. For example, in our polynomial:

• The zero -4 has a multiplicity of 1, which means it factors in as (x + 4)^1. The graph will just cross the x-axis at this point.
• The zero 0 has a multiplicity of 3, which means it factors in as (x)^3. At this zero, the graph will touch and bounce off the x-axis. This happens because an odd multiplicity greater than 1 causes the graph to flatten out before turning back, creating a distinct 'bounce'.
• The zero 2 has a multiplicity of 1, factored in as (x - 2)^1. Like the zero -4, it will simply cross the x-axis at this point.

Understanding multiplicity helps in sketching the graph of the polynomial accurately and predicting its behavior at different points.

Leading Coefficient

The leading coefficient of a polynomial is the coefficient of the term with the highest degree. It greatly influences the shape and orientation of the polynomial's graph. In our exercise:

• Initially, we don't know the leading coefficient, represented as 'a'.
• By substituting the given point (-2, 64) into the polynomial, we solve for 'a'.
• After simplification, we find that 'a' = -1.

The value of 'a' impacts whether the polynomial opens upwards or downwards. Since 'a' is -1, the polynomial graph opens downwards, indicative of an inverted shape for higher degree terms.

Degree of Polynomial

The degree of a polynomial is the highest power of x in the polynomial. It tells us about the general shape of the polynomial's graph and the maximum number of real zeros it can have. For our polynomial:

• With zeros -4, 0, and 2, and their respective multiplicities, the polynomial has a calculated degree of 5: (1 + 3 + 1).
• This degree fits our given condition for the polynomial.
• The degree also signifies that the polynomial can have at most 5 points where it crosses or touches the x-axis.

Higher degree polynomials tend to have more complex shapes, exhibiting multiple turns and different behavior at each critical point or real zero.