Problem 25 - College Algebra

Question

Find the zeros for each polynomial function and give the multiplicity for each zera. State whether the graph crosses the \(x\) -axis, or touches the \(x\) -axis and turns around, at each zera $$ f(x)=2(x-5)(x+4)^{2} $$

Step-by-Step Solution

Answer

The zeros of the function are \(x=5\) with multiplicity 1 and \(x=-4\) with multiplicity 2. The graph crosses the \(x\)-axis at \(x=5\), and touches the \(x\)-axis at \(x=-4\) and turns around.

Step 1: Find Zeros of the Function

Start by setting the function equal to zero to find the zeros or roots of the function. Here the function could be set as \[2(x-5)(x+4)^{2}=0\]. After setting both factors equal to zero, this gives two potential zeros: \(x=5\) and \(x=-4\).

Step 2: Find the Multiplicity of Each Zero

The multiplicity of a zero is determined by the exponent on its corresponding factor. For \(x=5\), the factor in the polynomial is \((x-5)\), and since there is no exponent, it is understood to be 1 indicating that \(x=5\) has multiplicity 1. For \(x=-4\), the factor in the polynomial is \((x+4)^{2}\) - the exponent, 2, indicates that \(x=-4\) has multiplicity 2.

Step 3: Determine if the Graph Crosses or Touches the x-Axis

Since x = 5 has a multiplicity of 1, which is an odd number, the graph crosses the \(x\)-axis at \(x=5\). Because \(x=-4\) has a multiplicity of 2, which is an even number, the graph will touch the \(x\)-axis at \(x=-4\) and turn around there.

Key Concepts

Zeros of a Polynomial

Zeros of a polynomial are the values of the variable that make the polynomial equal to zero. These are also known as the roots of the polynomial. Finding these values helps us understand the behavior of the polynomial function. To determine the zeros, we set the polynomial equation equal to zero. For example, in the polynomial function \( f(x) = 2(x-5)(x+4)^2 \), we find the zeros by solving \( 2(x-5)(x+4)^2 = 0 \). This equation can be broken down to two factors, \( (x-5) \) and \( (x+4)^2 \). Setting each factor equal to zero results in the zeros \( x = 5 \) and \( x = -4 \). Thus, these are the points on the graph where the polynomial touches or crosses the x-axis. It's essential to identify these points as they mark critical points in the graph's development.

Multiplicity

Multiplicity is a concept that tells us how many times a particular zero is repeated in a polynomial function. Essentially, it is defined by the power or exponent of the corresponding factor in the polynomial. For instance, in the polynomial \( f(x) = 2(x-5)(x+4)^2 \), the zero \( x = 5 \) comes from the factor \( (x-5) \). Since this factor does not have an exponent being displayed, this zero has a multiplicity of 1. This means that the zero occurs only once.On the other hand, the zero \( x = -4 \) emerges from the factor \( (x+4)^2 \), with the exponent of 2 indicating that its multiplicity is 2. This means \( x = -4 \) occurs twice or is repeated in the polynomial. Understanding multiplicity is crucial because it affects the shape of the graph near the zero, which we will explore next.

Graph Behavior at Zeros

The graph of a polynomial function behaves differently at each zero depending on the multiplicity of that zero. This behavior is observable at the x-axis where the polynomial either crosses or touches and turns around. If a zero has an odd multiplicity, the graph will cross the x-axis at that point. For our example with the zero \( x = 5 \), because its multiplicity is 1 (an odd number), the graph crosses the x-axis at \( x = 5 \).In contrast, if a zero has an even multiplicity, the graph will merely touch the x-axis at that point and turn around instead of crossing it. This is what happens with the zero \( x = -4 \) in the polynomial \( f(x) = 2(x-5)(x+4)^2 \). Here, the zero has a multiplicity of 2 (even number), so at \( x = -4 \), the graph touches the x-axis and reverses direction.Understanding how a polynomial’s graph behaves at its zeros is key in sketching the graph and predicting how it will pass through different regions on a graph.