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Chapter 2: Problem 5

Fill in the blanks. When a real zero of a polynomial function is of even multiplicity, the graph of \(f\) __________ the \(x\)-axis at \(x=a,\) and when it is of odd multiplicity, the graph of \(f\) __________ the \(x\)-axis at \(x=a\).

Short Answer

Expert verified
For even multiplicity, the graph of \(f\) touches the x-axis at \(x=a\), and for odd multiplicity, the graph crosses the x-axis at \(x=a\).

Step by step solution

01

Understanding Even Multiplicity

When a real zero (root) of a polynomial function has even multiplicity (e.g., 2, 4, 6, etc.), the graph will touch and bounce back from the x-axis at the point \(x = a\). This is because for even multiplicities, the polynomial function behaves similarly on either side of the x-axis.
02

Understanding Odd Multiplicity

When a real zero (root) of a polynomial function has odd multiplicity (e.g., 1, 3, 5, etc.), the graph will cross the x-axis at the point \(x = a\). This is because for odd multiplicities, the polynomial function behaves differently on either side of the x-axis.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Real Zeros
Real zeros of a polynomial function are the values of \(x\) that make the function equal to zero. When you plug a real zero into the polynomial, the result is zero, meaning the point \((x, 0)\) is on the graph.
Understanding real zeros is crucial because they depict where the graph intersects the x-axis.
  • To find the real zeros, solve the equation \(f(x) = 0\).
  • Real zeros can be found using techniques like factoring, using the quadratic formula, or applying synthetic division.
  • Once found, each real zero contributes to the shape and behavior of the polynomial鈥檚 graph.
Grasping the nature of real zeros aids not only in sketching the graph but also in understanding how the function behaves around these points. Thus, they are key in analyzing polynomial functions.
Even Multiplicity
Even multiplicity in polynomial zeros refers to zeros that repeat an even number of times. When we say a zero has an "even multiplicity," we mean the factor appears twice, four times, and so on in the factorization of the polynomial.
  • Example: In the polynomial \((x - 2)^4\), the zero \(x = 2\) has a multiplicity of 4, which is even.
  • When the graph reaches an x-intercept at a zero with even multiplicity, it touches the x-axis and "bounces" off it.
  • This touching and bouncing is akin to a ball dropping to the ground and returning upwards, indicating that the function does not change sign.
Such behavior results from the polynomial maintaining the same direction on either side of the zero. Recognizing even multiplicity helps predict turning points and the overall shape of the graph.
Odd Multiplicity
Odd multiplicity is when a zero occurs an odd number of times within a polynomial function, appearing once, thrice, etc. If a zero鈥檚 multiplicity is odd, it impacts how the polynomial graph behaves as it passes through the x-axis.
  • Example: In the polynomial \((x + 1)^3\), the zero \(x = -1\) has a multiplicity of 3, which is odd.
  • The graph crosses the x-axis when it encounters a zero of odd multiplicity. It advances from positive to negative y-values, or vice versa.
  • This crossing indicates that the function changes its sign upon passing the zero.
Comprehending odd multiplicities is essential in understanding how their presence modifies the overall trajectory of a graph. It allows for anticipating where the graph intersects the x-axis and how it continues beyond those intersections.

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Most popular questions from this chapter

Decide whether the statement is true or false. Justify your answer. It is possible for a third-degree polynomial function with integer coefficients to have no real zeros.

A small theater has a seating capacity of \(2000 .\) When the ticket price is \(\$ 20\) attendance is \(1500 .\) For each \(\$ 1\) decrease in price, attendance increases by 100 (a) Write the revenue \(R\) of the theater as a function of ticket price \(x\) (b) What ticket price will yield a maximum revenue? What is the maximum revenue?

Geometry You want to make an open box from a rectangular piece of material, 15 centimeters by 9 centimeters, by cutting equal squares from the corners and turning up the sides. (a) Let \(x\) represent the side length of each of the squares removed. Draw a diagram showing the squares removed from the original piece of material and the resulting dimensions of the open box. (b) Use the diagram to write the volume \(V\) of the box as a function of \(x .\) Determine the domain of the function. (c) Sketch the graph of the function and approximate the dimensions of the box that will yield a maximum volume. (d) Find values of \(x\) such that \(V=56 .\) Which of these values is a physical impossibility in the construction of the box? Explain.

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