Problem 31 - College Algebra

Question

In Exercises \(27-34,\) find the zeros for each polynomial function and give the multiplicity for each zero. State whether the graph crosses the \(x\) -axis, or touches the \(x\) -axis and turns around, at each zero. $$f(x)=x^{3}-2 x^{2}+x$$

Step-by-Step Solution

Answer

The solutions, or zeros, to the function \(f(x)=x^{3}-2 x^{2}+x\) are \(x=0\) and \(x=1\), each with a multiplicity of 1. The graph of the function crosses the x-axis at \(x=0\) and \(x=1\).

Step 1: Polynomial Simplification

The given polynomial \(f(x)=x^{3}-2 x^{2}+x\) can be simplified as \(f(x)= x(x^{2} - 2x +1)\). This simplifies the process of finding the roots as it breaks down the cubic polynomial into simpler components.

Step 2: Find the Zeros of the Function

To find the zeros, set the simplified polynomial equal to zero and solve for \(x\). This gives the equations: \(x=0\) and \(x^{2} - 2x +1=0\). From \(x=0\), we get one zero as 0. The quadratic equation \(x^{2} - 2x +1=0\) can further be solved to get the other roots. This quadratic equation is also a perfect square which yields \(x=1\) as the root.

Step 3: Determine the Multiplicity and Describe the Graph Behavior

The root \(x=0\) and \(x=1\) each appear only once, which means the multiplicity of the roots 0 and 1 is 1. Because of the odd multiplicity, the graph crosses the x-axis at \(x=0\) and \(x=1\).

Key Concepts

Polynomial Simplification

Understanding polynomial simplification is key to mastering algebra and pre-calculus problems. Simplifying polynomials often involves factoring, which breaks down more complicated polynomial expressions into products of simpler polynomials.

For example, take the function given in the exercise, f(x) = x^3 - 2x^2 + x. We can simplify this cubic polynomial by factoring out the greatest common factor (GCF), which is x in this case. This results in f(x) = x(x^2 - 2x + 1). Such simplification transforms the problem from dealing with a cubic polynomial to working with a linear term and a quadratic term.

This step is crucial because it lays the foundation for efficiently finding zeros, understanding the function's behavior, and graphing the polynomial. Factoring, in essence, makes an intricate problem far more manageable.

Root Multiplicity

When it comes to polynomials, the concept of 'root multiplicity' refers to how many times a particular solution (zero) appears. The root's multiplicity has profound implications on the graph of the polynomial function.

In our exercise, we found the zeros of the function f(x) to be at x = 0 and x = 1. Both these roots have multiplicity of 1, indicating they are simple roots. When a root's multiplicity is odd and equals 1, the graph intersects the x-axis at that point and continues on the opposite side.

It's important to recognize that zeros with higher multiplicities, especially even multiplicities, behave differently. Zeros with even multiplicities cause the graph to touch the x-axis and turn around without crossing, giving a different visual cue and implications for calculus-related concepts such as concavity and inflection points.

Graph Behavior at Zeros

Analyzing the behavior of a polynomial graph near its zeros provides insights into the nature of its roots. For our function, f(x) = x(x^2 - 2x + 1), we have zeros at x = 0 and x = 1 with multiplicity 1.

Our polynomial function crosses the x-axis at these zeros. Specifically, each time a graph crosses the x-axis at a zero, it indicates that the zero has an odd multiplicity. When a graph only touches the x-axis and turns around, it suggests that the zero has an even multiplicity.

This behavior helps in sketching rough graphs of polynomials without precise calculations and provides an early hint about the function's turning points—all this without the need for a graphing calculator, showcasing the power of algebraic understanding.