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Chapter 2: Problem 39

Find a polynomial (there are many) of minimum degree that has the given zeros. $$-5,-3,0,2,6$$

Short Answer

Expert verified
The polynomial is \( (x + 5)(x + 3)x(x - 2)(x - 6) \).

Step by step solution

01

Understand the Zeros Given

The problem provides the zeros of the polynomial as \(-5, -3, 0, 2,\) and \(6\). Each zero corresponds to a factor of the polynomial in the form \((x - \text{zero})\). For example, if the zero is \(-5\), the factor is \((x + 5)\) since \(x - (-5) = x + 5\).
02

Form Factors from Zeros

Using the zeros, we form the corresponding factors of the polynomial:- For \(-5\), the factor is \((x + 5)\).- For \(-3\), the factor is \((x + 3)\).- For \(0\), the factor is \(x\).- For \(2\), the factor is \((x - 2)\).- For \(6\), the factor is \((x - 6)\).
03

Multiply the Factors to Form the Polynomial

The minimum-degree polynomial that has these zeros is the product of these factors:\[ P(x) = (x + 5)(x + 3)x(x - 2)(x - 6) \]This polynomial contains each zero as a root exactly once, resulting in a polynomial with a degree equal to the number of unique zeros, which is 5 in this case.

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Key Concepts

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

Factors of a polynomial
When finding the factors of a polynomial, think about breaking it down into simpler pieces. Each zero of a polynomial represents a point where the polynomial touches or crosses the x-axis.
If a zero is given as part of the problem, translate it into a factor by using the form \((x - \text{zero})\). For example:
  • A zero of \-5 leads to the factor \((x + 5)\) since \(x - (-5) = x + 5\).
  • Similarly, a zero of \0 becomes simply \(x\) because the expression is \(x - 0 = x\).

The factors are critical steps in forming your polynomial because multiplying them together gives you the polynomial equation.
Remember, the more zeros you have, the more factors you'll need to multiply together.
Degree of a polynomial
The degree of a polynomial is a key concept that describes the highest power of the variable within the polynomial.
In simpler terms, it's the largest exponent you will see if the polynomial is fully expanded.

For example, suppose we have a polynomial formed by multiplying the factors from our problem: \((x + 5)(x + 3)x(x - 2)(x - 6)\).
  • Each variable \(x\) in the factors contributes to the degree.
  • When there are five different factors each with an \(x\) term, the degree of this polynomial is \(5\).

Thus, the lowest-degree polynomial that accommodates all these zeros is a polynomial of degree 5.
It's important to know that the degree also dictates the number of roots and the shape of the graph.
Zero of a function
A zero of a function is essentially a solution to the equation where the function equals zero.
In other words, it's the x-value(s) where the function touches or crosses the x-axis.

Zeros are important because finding them helps us understand the function's behavior and graph.
  • For instance, the zeros \(-5, -3, 0, 2,\) and \(6\) give us some precise points on the graph of the polynomial \(P(x)\).
  • Each zero helps construct the polynomial and ensures the polynomial crosses or touches the x-axis at that point.

Understanding the zero of a function is integral in solving polynomial equations, as it highlights the critical points of intersection or tangency with the x-axis.
Roots of equations
The roots of equations are synonymous with the zeros of a polynomial, referring to the values of \(x\) that make the entire polynomial equal to zero.
These roots provide insight into where and how the graph of the polynomial interacts with the x-axis.

In the context of our example, the roots \(-5, -3, 0, 2,\) and \(6\) are where the polynomial \(P(x)\) equals zero, specifically:
  • Each root satisfies the equation \((x - r) = 0\), where \(r\) is a specified root.
  • For example, if \(r = -5\), then \((x + 5) = 0\), marking \(-5\) as a root.

By identifying all roots, we can predict how the function behaves across different intervals on the graph.
Finding the roots or zeros is crucial in fully understanding and solving equations or inequalities involving polynomials.

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Most popular questions from this chapter

(a) Identify all asymptotes for each function. (b) Plot \(f(x)\) and \(g(x)\) in the same window. How does the end behavior of the function \(f\) differ from that of \(g ?\) (c) Plot \(g(x)\) and \(h(x)\) in the same window. How does the end behavior of \(g\) differ from that of \(h ?\) (d) Combine the two expressions into a single rational expression for the functions \(g\) and \(h .\) Does the strategy of finding horizontal and slant asymptotes agree with your findings in (b) and (c)? $$\begin{aligned} &f(x)=\frac{2 x}{x^{2}-1}, g(x)=x+\frac{2 x}{x^{2}-1}, \text { and }\\\ &h(x)=x-3+\frac{2 x}{x^{2}-1} \end{aligned}$$

In Exercises \(35-44,\) graph the quadratic function. $$f(x)=\frac{1}{2} x^{2}-\frac{1}{2}$$

In Exercises \(83-86,\) explain the mistake that is made. There may be a single mistake or there may be more than one mistake. Rewrite the following quadratic function in standard form: $$ f(x)=-x^{2}+2 x+3 $$ Solution: Step 1: Group the variables together. \(\quad\left(-x^{2}+2 x\right)+3\) Step 2: Factor out a negative. \(-\left(x^{2}+2 x\right)+3\) Step 3: Add and subtract 1 inside the parentheses. \(\quad-\left(x^{2}+2 x+1-1\right)+3\) Step 4: Factor out the \(-1 . \quad-\left(x^{2}+2 x+1\right)+1+3\) Step 5: Simplify. \(-(x+1)^{2}+4\) This is incorrect. What mistake(s) was made?

In Exercises \(87-90\), determine whether each statement is true or false. A quadratic function must have a \(y\) -intercept.

A financial institution offers to its investors a variable annual interest rate using the formula $$r(x)=\frac{4 x^{2}}{x^{2}+2 x+5}$$ where \(x\) is the amount invested in thousands of dollars. a. What is the annual interest rate for an investment of $$ 8,000 ?\( b. What is the annual interest rate for an investment of $$ 20,000 ?\) c. What is the maximum annual interest rate offered by them?
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