Problem 27 - 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. $$f(x)=2(x-5)(x+4)^{2}$$

Step-by-Step Solution

Answer

The zeros of the function f(x) are 5 and -4. 5 has multiplicity 1 and at this zero, the graph crosses the x-axis. -4 has multiplicity 2 and at this zero, the graph touches the x-axis and turns around.

Step 1: Identify the Zeros

To find the zeros of a function, we need to set the function equal to zero and solve. So, in this case, we set \(f(x) = 0\). This gives us the equation \(2(x - 5)(x + 4)^2 = 0\). By setting each factor equal to zero, we find \(x - 5 = 0\) and \(x + 4 = 0\). Solving these equations for \(x\), we get \(x = 5\) and \(x = -4\). So, the zeros of this function are 5 and -4.

Step 2: Determine the Multiplicity

The multiplicity of a zero is given by the exponent attached to the corresponding factor in the factored form of the function. From the given function, we can see that the zero 5 has an attached exponent of 1, which means it has a multiplicity of 1. For the zero -4, the attached exponent is 2, so it has a multiplicity of 2.

Step 3: Determine Behavior at Zeros

The behavior of the function's graph at a zero is determined by the multiplicity of the zero. If the multiplicity is odd, the graph crosses the x-axis at the zero. If the multiplicity is even, the graph touches the x-axis at the zero and turns around. As found in Step 2, 5 has a multiplicity of 1 (odd), so the graph crosses the x-axis at \(x = 5\). -4 has a multiplicity of 2 (even), so the graph touches the x-axis at \(x = -4\) and turns around.

Key Concepts

Polynomial Functions

A polynomial function is a mathematical expression that consists of variables raised to whole-number exponents, usually in the form of a sum. It looks like this: \[ f(x) = a_n x^n + a_{n-1} x^{n-1} + \ + a_1 x + a_0 \] where \(a_n, a_{n-1}, \ldots, a_0\) are constants, and \(n\) represents the degree of the polynomial. In our exercise, the function \(f(x) = 2(x-5)(x+4)^2\) represents a polynomial which, when expanded, becomes a degree three polynomial (once you multiply the factors). This is because the product of the terms determines the highest power of \(x\).

• Each term in a polynomial function contributes to the shape of its graph, its zeros, and its intersections with the axes.
• The real challenge lies in finding these characteristics, particularly the zeros, and understanding their implications.

Remember, when dealing with polynomial functions, your first step is often to factor them so you can better understand their zeros and behaviors.

Multiplicity of Zeros

The multiplicity of a zero refers to the number of times a particular zero occurs in the polynomial. This is determined by the exponent of the corresponding factor in the factored form of the polynomial. Understanding the multiplicity is important because it dictates how the graph interacts with the \(x\)-axis at those points. For our particular polynomial:

• Zero 5 comes from the factor \((x-5)\), with an exponent of 1. This means 5 is a zero with multiplicity 1.
• Zero -4 comes from the factor \((x+4)^2\), indicating -4 is a zero with multiplicity 2.

When the multiplicity is:

• Odd: The graph will cross the \(x\)-axis.
• Even: The graph will touch and turn around at the zero.

This knowledge helps predict the shape and turn points of the polynomial graph.

Graph Behavior at Zeros

Understanding the behavior of a polynomial function's graph at its zeros gives us insights into how the function changes over its domain. Graph behavior is intricately linked to the multiplicity of zeros:

• Zeros with on odd multiplicity, such as 5 in our function, mean the graph will cross the \(x\)-axis at that zero.
• Zeros with an even multiplicity, like -4 in our function, imply the graph will touch the \(x\)-axis and turn back, creating a flat spot or a "bounce".

Why does this matter? It helps in sketching graphs and predicting how the function behaves without detailed calculations. By merely examining the multiplicities of zeros, we can expect:

• Crossing at points of odd multiplicity, showing where the function changes sign.
• Tangency at points of even multiplicity, indicating potential minima or maxima in that section of the graph.

These crucial observations aid students and professionals alike in mapping out polynomial graphs efficiently.