Problem 52 - Precalculus with Calculus Previews

Question

Consider the polynomial function \(f(x)=(x-5) ^{2 m}(x+1)^{2 n-1}\), where \(m\) and \(n\) are positive integers. (a) For what values of \(m\) does the graph of \(f\) cross the \(x\) -axis at \((5,0) ?\) (b) For what values of \(n\) does the graph of \(f\) cross the \(x\) -axis at \((-1,0) ?\)

Step-by-Step Solution

Answer

(a) No values of m; (b) All positive integers n.

Step 1: Analyze the Crossing Condition

For a graph to cross the x-axis at a point \((x, 0)\), the zero at that point must yield an odd power in the polynomial. This means the factor multiplicity should be odd for the crossing to occur.

Step 2: Analyze the Polynomial at x=5

The factor \((x-5)\) in the polynomial is raised to the power of \(2m\). In order for the graph to cross the x-axis at \((5,0)\), the exponent, \(2m\), must be odd. However, 2m is always even for any integer \(m\), so the graph cannot cross at \((5,0)\) for any positive integer \(m\).

Step 3: Analyze the Polynomial at x=-1

The factor \((x+1)\) in the polynomial is raised to the power of \(2n-1\). For the graph to cross the x-axis at \((-1,0)\), the exponent \(2n-1\) must be odd. This is always the case for any positive integer \(n\), as subtracting 1 from an even number results in an odd number. Thus, the graph will cross at \((-1,0)\) for all positive integers \(n\).

Step 4: Conclude the Values

Summarize the findings: the graph does not cross the x-axis at \((5,0)\) for any positive integer \(m\), and it does cross at \((-1,0)\) for all positive integers \(n\).

Key Concepts

Factor Multiplicity

Understanding factor multiplicity is crucial in analyzing polynomial functions. When a polynomial is expressed in its factored form like \( f(x) = (x-5)^{2m}(x+1)^{2n-1} \), the multiplicity of each factor provides valuable insight into the behavior of the function's graph at its zeroes.

• A factor's multiplicity refers to the number of times it appears in the polynomial.
• If a factor, say \((x-a)\), appears with an odd multiplicity, the graph will intersect the x-axis at \(x = a\).
• Conversely, if the multiplicity is even, the graph will merely touch and then reflect away from the x-axis at that zero, without crossing it.

This distinction is essential in determining where and how a graph behaves in specific intervals. For instance, in our original problem, the factor \((x-5)^{2m}\), where \(2m\) is always even, implies the graph touches but doesn’t cross the x-axis at \(x=5\). Understanding this allows us to make predictions about the graph's behavior at its zeroes.

Zeroes of Polynomials

The zeroes of a polynomial are the x-values where the polynomial evaluates to zero. These points are crucial as they often tell you where the graph of the polynomial will interact with the x-axis.

• In our example, the polynomial function \(f(x) = (x-5)^{2m}(x+1)^{2n-1}\) has two factors that indicate potential zeroes: \(x-5\) and \(x+1\).
• The zero \(x=5\) arises from \((x-5)^{2m}\) and is influenced by the factor's even multiplicity, which means the graph touches the x-axis at this point without crossing it.
• The zero \(x=-1\) is derived from \((x+1)^{2n-1}\). Here, since \(2n-1\) is odd, it ensures the graph crosses the x-axis at \((-1,0)\) for any positive \(n\).

Comprehending the concept of zeroes and how their multiplicities affect their interaction with the x-axis is key to predicting and understanding the graphical behavior of polynomial functions.

Graph of a Polynomial Function

The graph of a polynomial function is a representation of the solutions of the polynomial equation, where the x-values (zeroes) highlight the interaction of the graph with the x-axis. Here's how polynomial graphs behave based on the factors:

• A zero of odd multiplicity, like \(x=-1\) from the factor \((x+1)^{2n-1}\), indicates that the graph crosses the x-axis at that point and changes direction.
• In contrast, a zero of even multiplicity, observed at \(x=5\) from \((x-5)^{2m}\), signifies the graph touches the x-axis and bounces back in the same direction it came from, creating a tangent.
• The degree of the polynomial, obtained by summing the multiplicities of all factors, influences the direction of the ends of the graph. The behavior is typically characterized by the leading term when expanded.

Understanding these principles allows you to sketch and anticipate the shape of the graph of polynomial functions. Predicting crossings and tangents on the x-axis helps conduct further analysis like determining intervals of increase and decrease.