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Chapter 3: Problem 6

Determine the number of zeros of the polynomial function. $$f(x)=3-7 x^{2}-5 x^{4}+9 x^{6}$$

Short Answer

Expert verified
The polynomial \(f(x)=3-7 x^{2}-5 x^{4}+9 x^{6}\) has 6 zeros.

Step by step solution

01

Identify the degree of the polynomial

The degree of the polynomial is identified as the highest power of x. In this case, the highest power of x is 6, so the degree of the polynomial is 6.
02

Apply the Fundamental Theorem of Algebra

The Fundamental Theorem of Algebra states that a polynomial of degree n has exactly n roots, or zeros, considering multiplicities and complex roots. Applying this to the current situation means that the given polynomial \(f(x)=3-7 x^{2}-5 x^{4}+9 x^{6}\) has exactly 6 zeros.

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Key Concepts

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

Degree of a Polynomial
In a polynomial function, the degree is the highest power of the variable that appears in the expression. This value is important as it signifies the fundamental characteristics of the polynomial. The degree determines the shape and behavior of the polynomial's graph, as well as its end behavior. To find the degree of a polynomial like \( f(x)=3-7x^{2}-5x^{4}+9x^{6} \), look for the term with the highest exponent. Here, the term \( 9x^{6} \) reveals that the highest degree is 6.
This means that the polynomial is a sixth-degree polynomial. Some key points to remember about degrees of polynomials are:
  • The degree indicates the maximum number of solutions, or zeros, the polynomial can have.
  • The degree also suggests how the graph of the polynomial behaves at extremes, either positive or negative infinity.
Fundamental Theorem of Algebra
The Fundamental Theorem of Algebra provides a critical insight into the structure of polynomial equations. Essentially, it tells us that any non-zero polynomial equation, with complex coefficients, has as many roots as its degree, when counted with multiplicity. For example, consider the polynomial \( f(x)=3-7x^{2}-5x^{4}+9x^{6} \).
According to the theorem, because this is a sixth-degree polynomial, it has exactly six zeros. These zeros could be real or complex numbers and some may occur more than once, this counting includes such multiplicities.
Here are some important aspects to understand:
  • The polynomial degree matches the number of roots the polynomial has, considering multiplicity.
  • Complex roots must appear in conjugate pairs, so if one complex number is a root, its conjugate must also be a root.
Zeros of a Polynomial
Zeros of a polynomial are the values of \( x \) that satisfy the equation \( f(x) = 0 \). Finding these zeros, also known as roots, is like solving a puzzle to find where the graph of the polynomial intersects the x-axis. The zeros are crucial because they provide insight into the properties and behavior of the polynomial function.
The number of zeros is determined by the degree of the polynomial. For example, since our polynomial \( f(x)=3-7x^{2}-5x^{4}+9x^{6} \) is of degree 6, it must have 6 zeros, including those repeated or complex. Understanding these zeros can help:
  • Factorize the polynomial or break it into smaller multiplicative parts.
  • Determine critical points for solving inequalities or optimization problems.
  • Provide a complete set of solutions for equations linked with the polynomial.
Knowing how to find and interpret these zeros is an essential skill in algebra and calculus.

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Most popular questions from this chapter

Describe the right-hand and left-hand behavior of the graph of the polynomial function. $$g(x)=6-4 x^{2}+x-3 x^{5}$$

Analyzing a Graph In Exercises \(47-58\), analyze the graph of the function algebraically and use the results to sketch the graph by hand. Then use a graphing utility to confirm your sketch. $$f(x)=\frac{1}{4} x^{4}-2 x^{2}$$

Comparing Graphs Use a graphing utility to graph the functions given by \(f(x)=x^{3}, g(x)=x^{5}\), and \(h(x)=x^{7} .\) Do the three functions have a common shape? Are their graphs identical? Why or why not?

Regression Problem Let \(x\) be the angle (in degrees) at which a baseball is hit with no spin at an initial speed of 40 meters per second and let \(d(x)\) be the distance (in meters) the ball travels. The table shows the distances for the different angles at which the ball is hit. (Source: The Physics of Sports) $$ \begin{aligned} &\begin{array}{|l|l|l|l|l|l|} \hline \text { Angle, } x & 10 & 15 & 30 & 36 & 42 \\ \hline \text { Distance, } d(x) & 58.3 & 79.7 & 126.9 & 136.6 & 140.6 \\ \hline \end{array}\\\ &\begin{array}{|l|l|l|l|l|l|} \hline \text { Angle, } x & 44 & 45 & 48 & 54 & 60 \\ \hline \text { Distance, } d(x) & 140.9 & 140.9 & 139.3 & 132.5 & 120.5 \\ \hline \end{array} \end{aligned} $$ (a) Use a graphing utility to create a scatter plot of the data. (b) Use the regression feature of a graphing utility to find a quadratic model for \(d(x)\). (c) Use a graphing utility to graph your model for \(d(x)\) with the scatter plot of the data. (d) Find the vertex of the graph of the model from part (c). Interpret its meaning in the context of the problem.

Analyzing a Graph In Exercises \(47-58\), analyze the graph of the function algebraically and use the results to sketch the graph by hand. Then use a graphing utility to confirm your sketch. $$f(x)=4 x^{2}-x^{3}$$
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