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Chapter 3: Problem 61

What are the zeros of a polynomial function and how are they found?

Short Answer

Expert verified
Zeros of a polynomial function are the x-values where the function equals zero. They are found by using various methods including isolation of terms, quadratic formula, synthetic division, or factoring, based on the degree of the polynomial.

Step by step solution

01

Understanding Zeros of Polynomial

The zeros of a polynomial are those values of the variable which make the polynomial equal to zero. In other words, if inserting a certain x-value into the equation makes the outcome zero, then that x-value is a zero of the polynomial.
02

Methods to Find Zeros

There are different methods to find the zeros depending on the degree of the polynomial. For linear polynomials (degree 1), just simplifying the equation to the term containing the variable isolated on one side will give the solution. For quadratic polynomials (degree 2), quadratic formula can be used. For higher degree polynomials, one might need to use synthetic division or factoring.
03

Applying Methods

Apply these methods depending on the degree of your polynomial function. If it is a simple linear equation, isolate the variable term to find the zero. If it is a quadratic equation, use the quadratic formula \(-\frac{b}{2a} \pm \sqrt{(\frac{b}{2a})^2 - a}\), where a, b, and c are coefficients in a quadratic equation in the form \(ax^2 + bx + c = 0\). For higher degree equations, use synthetic division or factoring, methods which generally require a bit more of computational work.

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Most popular questions from this chapter

In Exercises \(55-56,\) use a graphing utility to determine upper and lower bounds for the zeros of \(f .\) Does synthetic division verify your observations? $$ f(x)=2 x^{3}+x^{2}-14 x-7 $$

In Exercises \(74-77\), use a graphing utility with a viewing rectangle large enough to show end behavior to graph each polynomial function. $$f(x)=x^{3}+13 x^{2}+10 x-4$$

In Exercises \(1-10\), determine which functions are polynomial functions. For those that are, identify the degree. $$f(x)=\frac{x^{2}+7}{3}$$

In Exercises \(1-10\), determine which functions are polynomial functions. For those that are, identify the degree. $$g(x)=7 x^{5}-\pi x^{3}+\frac{1}{5} x$$

In Exercises \(35-50\) a. Use the Leading Coefficient Test to determine the graphs end behavior. b. Find \(x\) -intercepts by setting \(f(x)=0\) and solving the resulting polynomial equation. State whether the graph crosses the \(x\)-axis, or touches the \(x\)-axis and turns around, at each intercept. c. Find the \(y\) -intercept by setting \(x\) equal to 0 and computing \(f(0)\) d. Determine whether the graph has \(y\) -axis symmetry, origin symmetry, or neither. e. If necessary, find a few additional points and graph the function. Use the fact that the maximum number of turning points of the graph is \(n-1\) to check whether it is drawn correctly. $$f(x)=x^{3}+2 x^{2}-x-2$$
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