Question

Find a polynomial function whose real zeros and degree are given. Answers will vary depending on the choice of the leading coefficient. $$ \text { Zeros: }-2, \text { multiplicity } 2 ; 4, \text { multiplicity } 1 ; \text { degree } 3 $$

Short answer

The polynomial function is \( a(x + 2)^2(x - 4) \).

Step-by-step solution

Identify the zeros and their multiplicities

Given the zeros and their multiplicities: -2 with multiplicity 2, and 4 with multiplicity 1. The degree of the polynomial is 3.

Construct factors from the zeros

Convert each zero into a corresponding factor. A zero at x = -2 with multiplicity 2 gives us \( (x + 2)^2 \). A zero at x = 4 with multiplicity 1 gives us \( (x - 4) \).

Form the polynomial

Combine the factors to form the polynomial: \(a(x + 2)^2(x - 4)\), where \( a \) is the leading coefficient.

Expand the polynomial

Expand \( (x + 2)^2 \) first: \( (x + 2)^2 = x^2 + 4x + 4 \). Then, multiply by \( (x - 4) \): \[ a(x^2 + 4x + 4)(x - 4) = a(x^3 - 4x^2 + 4x^2 - 16x + 4x - 16) = a(x^3 - 12x - 16) \].

Key concepts

Real Zeros

In a polynomial function, the 'real zeros' are the values of x for which the polynomial equals zero. These zeros are solutions to the equation formed by setting the polynomial equal to zero. For example, in the given polynomial, we have real zeros at -2 and 4.
Real zeros are crucial because they help us understand where the polynomial crosses the x-axis. They can be found using various methods such as factoring, using the Rational Root Theorem, or through numerical methods like the Newton-Raphson method.
Knowing the real zeros of a polynomial helps in graphing and analyzing the function's behavior.

Multiplicity

The 'multiplicity' of a zero refers to how many times that zero occurs as a root of the polynomial. For example, a zero at x = -2 with multiplicity 2 means that (x + 2) is squared in the polynomial.
Multiplicity affects the shape of the polynomial graph. If the multiplicity is odd, the graph will cross the x-axis at that zero. If the multiplicity is even, the graph will touch the x-axis and turn around at that zero.
In our example, the zero at -2 has a multiplicity of 2, causing the graph to touch and bounce back at -2. The zero at 4 has a multiplicity of 1, making the graph cross the x-axis at 4.

Degree of Polynomial

The 'degree of a polynomial' is the highest exponent of the variable x in the polynomial. It tells us the maximum number of times the graph of the polynomial can intersect the x-axis and the overall shape of the polynomial's graph.
For instance, the given polynomial in our problem has a degree of 3 because the highest exponent of x is 3 in the term x^3.
Higher degree polynomials have more complex graphs with more turns. A polynomial of degree n will have at most n-1 turning points and n-1 zero crossings. Understanding the polynomial's degree helps in predicting its end behavior and the general layout of its graph.

Leading Coefficient

The 'leading coefficient' is the coefficient of the term with the highest degree in a polynomial. It greatly influences the end behavior of the polynomial's graph.
In our polynomial, the leading coefficient is denoted by the variable a, and it multiplies the highest degree term x^3.
If the leading coefficient is positive, the polynomial's ends will rise to infinity. If it's negative, the ends will fall to negative infinity. Leading coefficients also affect the vertical stretch or compression of the polynomial's graph. In our problem, since we didn't specify a value for a, the polynomial could vary depending on the choice.