Problem 29 - Algebra and Trigonometry

Question

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 zeros of the function \(f(x) = x^{3}-2 x^{2}+x\) are \(x = 0\) (with multiplicity 1) and \(x = 1\) (with multiplicity 2). The graph crosses the x-axis at \(x = 0\) and touches the x-axis and turns around at \(x = 1\).

Step 1: Setting the Function Equal to Zero

Firstly, set the function \(f(x) = x^{3}-2 x^{2}+x\) equal to zero. So the equation will be: \(x^{3}-2 x^{2}+x = 0\).

Step 2: Factoring out the Common Factor

In the equation, \(x\) is the common factor for all terms. Factor out this common factor and the equation becomes: \(x( x^{2}-2 x+1) = 0\).

Step 3: Factoring the Quadratic

Next step is factoring the quadratic expression in the parentheses. This quadratic is a perfect square trinomial which can be factored as follows: \(x( x-1)^{2} = 0\).

Step 4: Solving for x

Now, set each factor equal to zero and solve for \(x\). Setting \(x = 0\) and \(x-1 = 0\), we get two solutions: \(x = 0\) and \(x = 1\). The solution \(x = 1\) is repeated because the factor appears twice inside the parentheses. Thus, it has a multiplicity of 2.

Step 5: Observing the Graph's Behavior at Each Zero

Since \(x = 0\) is a root of multiplicity 1 (odd), the graph crosses the x-axis at \(x = 0\). On the other hand, as \(x = 1\) is a root of multiplicity 2 (even), the graph touches the x-axis at \(x = 1\) and turns around.

Key Concepts

Factoring Polynomials

Factoring polynomials is a crucial step in finding the zeros of a polynomial function. When you factor a polynomial, you're breaking it down into simpler terms—called factors—that can be multiplied together to get the original polynomial. In our example, the polynomial function is given as \(f(x) = x^{3} - 2x^{2} + x\).

The first step is recognizing that each term in the polynomial share a common factor. In this case, \(x\) is the common factor. By factoring out \(x\), the expression simplifies to \(x(x^{2} - 2x + 1) = 0\).

Next, we factor the quadratic \(x^{2} - 2x + 1\), which is a perfect square trinomial. It can be rewritten as \((x-1)^2\). Therefore, the factorization of the polynomial becomes \(x(x-1)^2 = 0\). Factoring is often necessary because it makes finding the roots, or zeros, simpler. Once the polynomial is fully factored, we can easily identify the zeros by setting each factor equal to zero.

Multiplicity of Zeros

The multiplicity of a zero refers to the number of times a particular solution occurs. It indicates how many times the graph of the function touches or crosses the x-axis at that point. In our problem, we identified the expression \(x(x-1)^2 = 0\) and found the zeros at \(x = 0\) and \(x = 1\).

• The zero at \(x = 0\) is of multiplicity 1. This means the zero occurs only once in the factorization and the graph intersects the x-axis at this point.
• The zero at \(x = 1\) has a multiplicity of 2 because the factor \((x-1)\) is squared. This indicates the zero appears twice, which affects how the graph behaves at that point.

Multiplicity is useful for predicting the graph's behavior at each zero. More times the zero appears, higher its multiplicity and more it influences the graph.

Graph Behavior at Zeros

Understanding the graph's behavior at zeros is essential when analyzing polynomial functions. Let's see how a graph typically behaves based on the multiplicity of its zeros.

When a zero has an odd multiplicity, like our zero at \(x = 0\), the graph will cross the x-axis. This means at \(x = 0\), the graph dips below and crosses above or vice versa over the x-axis. This crossing happens because odd multiplicity zeros imply a change in the sign of the function values on either side of the zero.

In contrast, when a zero has an even multiplicity, such as \(x = 1\) in our exercise, the graph behaves differently. Here, the polynomial function touches the x-axis but instead of crossing it, it turns back. At \(x = 1\), the graph "bounces off" the x-axis, which means there's no change in sign for function values on either side of the zero.

Predicting how the graph behaves at its zeros helps in creating an accurate sketch of the function or understanding its visual representation. This knowledge supplements the process of finding and understanding zeros in polynomial functions.