Problem 20 - Algebra for College Students

Question

For Problems \(1-22\), graph each of the polynomial functions. $$ f(x)=(x-1)^{2}(x+2) $$

Step-by-Step Solution

Answer

Graph touches the x-axis at \( x = 1 \), crosses at \( x = -2 \), and demonstrates cubic end behavior.

Step 1: Identify the Polynomial Function

The function given is \( f(x) = (x-1)^2(x+2) \). This is a polynomial function of degree 3 due to the multiplication of the terms \( (x-1)^2 \) and \( (x+2) \).

Step 2: Determine the Zeros and Their Multiplicities

To find the zeros, set \( f(x) = 0 \): \( (x-1)^2(x+2) = 0 \). The zeros are \( x=1 \) with multiplicity 2, and \( x=-2 \) with multiplicity 1. This means the graph will touch the x-axis at \( x=1 \) and cross it at \( x=-2 \).

Step 3: Determine the End Behavior

Since it is a cubic function (degree 3), the end behavior will follow that of \( x^3 \). As \( x \to \infty \), \( f(x) \to \infty \), and as \( x \to -\infty \), \( f(x) \to -\infty \).

Step 4: Plot the Key Points and Sketch

Mark the zeros on the graph: touch at \( x=1 \) and cross at \( x=-2 \). Plot additional points around these values to assist with accuracy. Ensure the graph shows proper end behavior and respects the multiplicity at each zero.

Key Concepts

Polynomial Function

A polynomial function is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. In simpler terms, it's an equation consisting of variables raised to whole number exponents. The function can be expressed in general as:

• The standard form: \( f(x) = a_nx^n + a_{n-1}x^{n-1} + \, \ldots \, + a_1x + a_0 \)
• Here, \(a_n, a_{n-1}, \, \ldots \, , a_0 \) are constants known as coefficients, and \(n\) is a non-negative integer called the degree of the polynomial.

In the given exercise, the polynomial function is \( f(x) = (x-1)^2(x+2) \). When expanded, this would be equivalent to a cubic polynomial since its highest degree term upon expansion is \( x^3 \). This determines both the shape and characteristics of the graph.

Zeros of Polynomial

The zeros of a polynomial are the values of \(x\) that make the polynomial equal to zero. These are also the points where the graph of the polynomial intersects or touches the x-axis.

To find the zeros for \( f(x) = (x-1)^2(x+2) \):

• Set the equation equal to zero: \( (x-1)^2(x+2) = 0 \).
• Solve for \(x\), which gives \( x = 1 \) and \( x = -2 \).

These solutions are the zeros of the polynomial. At \( x = 1 \), due to the squared term \((x-1)^2\), the graph will "bounce" or touch the x-axis without crossing. At \( x = -2 \), the graph will cross the x-axis because the term \((x+2)\) is linear and not repeated.

End Behavior

The end behavior of a polynomial function describes how the function behaves as the input \(x\) approaches positive or negative infinity.

• For \( f(x) = (x-1)^2(x+2) \), the degree is 3, which is odd.
• The leading term upon expansion is \( x^3 \), indicating that the polynomial will mimic the behavior of \( x^3 \).

For odd-degree polynomial functions like cubics:

• As \( x \to \infty \), \( f(x) \to \infty \).
• As \( x \to -\infty \), \( f(x) \to -\infty \).

This describes the ends of the graph moving in opposite directions, upwards to the right and downwards to the left.

Multiplicity of Roots

The multiplicity of a root refers to the number of times it is repeated as a factor in the polynomial.

• If the root is repeated, its multiplicity is greater than 1.
• Each factor raised to a power in the polynomial contributes to the multiplicity.

In \( f(x) = (x-1)^2(x+2) \):

• The root \( x=1 \) occurs twice because of \((x-1)^2\) giving it a multiplicity of 2.
• This means the graph touches the x-axis at this point and doesn't cross.
• The root \( x=-2 \) has multiplicity 1, indicating it crosses the x-axis there.

Understanding root multiplicity helps predict the behavior of the graph at each zero, whether it bounces off or pierces through the x-axis.