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Chapter 3: Problem 6

List all possible rational zeros given by the Rational Zeros Theorem (but don't check to see which actually are zeros). $$U(x)=12 x^{5}+6 x^{3}-2 x-8$$

Short Answer

Expert verified
Possible rational zeros are \( \pm 1, \pm \frac{1}{2}, \pm \frac{1}{3}, \pm \frac{1}{4}, \pm \frac{1}{6}, \pm \frac{1}{12}, \pm 2, \pm \frac{2}{3}, \pm 4, \pm \frac{4}{3}, \pm 8, \pm \frac{8}{3} \).

Step by step solution

01

Identify Constant Term and Leading Coefficient

For the polynomial \(U(x) = 12x^5 + 6x^3 - 2x - 8\), the constant term \(a_0\) is \(-8\) and the leading coefficient \(a_n\) is \(12\).
02

List Factors of Constant Term

List all positive and negative factors of \(-8\). They are: \( \pm 1, \pm 2, \pm 4, \pm 8\).
03

List Factors of Leading Coefficient

List all positive and negative factors of \(12\). They are: \( \pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12\).
04

Apply the Rational Zeros Theorem

Combine these factors to form all possible rational zeros. The possible rational zeros are given by \( \pm \frac{p}{q} \), where \(p\) are the factors of \(-8\) and \(q\) are the factors of \(12\).
05

List All Possible Rational Zeros

Calculate all combinations of \(\frac{p}{q}\) resulting in the possible rational zeros: \( \pm 1, \pm \frac{1}{2}, \pm \frac{1}{3}, \pm \frac{1}{4}, \pm \frac{1}{6}, \pm \frac{1}{12}, \pm 2, \pm \frac{2}{3}, \pm \frac{2}{4}, \pm \frac{2}{6}, \pm \frac{2}{12}, \pm 4, \pm \frac{4}{3}, \pm \frac{4}{6}, \pm 8, \pm \frac{8}{3}, \pm \frac{8}{4}, \pm \frac{8}{6}\).
06

Simplify

Reduce the fractions to list distinct possible rational zeros. The final list of possible rational zeros is: \(\pm 1, \pm \frac{1}{2}, \pm \frac{1}{3}, \pm \frac{1}{4}, \pm \frac{1}{6}, \pm \frac{1}{12}, \pm 2, \pm \frac{2}{3}, \pm \frac{1}{2}, \pm 4, \pm \frac{4}{3}, \pm 8, \pm \frac{8}{3}\).

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Key Concepts

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

polynomial functions
Polynomial functions are mathematical expressions that involve variables raised to whole number powers, combined using addition, subtraction, and multiplication. They are of the form:
  • A polynomial in one variable is expressed as:\[ax^n + bx^{n-1} + cx^{n-2} + \,\ldots\, + zx^0\]where \(a, b, c,\) and \(z\) are constants and \(n\) is a non-negative integer.
The example polynomial function given is \(U(x) = 12x^5 + 6x^3 - 2x - 8\). In this polynomial:
  • The highest power is 5, making it a polynomial of degree 5.
  • The term \(12x^5\) is the leading term because it contains the highest power of \(x\).
  • The constant term is the part of the polynomial that does not contain the variable \(x\), which in this case is \(-8\).
Polynomial functions are used to model a wide range of real-world situations, from physics to finance.
leading coefficient
The leading coefficient is the coefficient of the term with the highest power in a polynomial. It plays a crucial role in determining the end behavior of the polynomial's graph:
  • In \(U(x) = 12x^5 + 6x^3 - 2x - 8\), the leading term is \(12x^5\).
  • The leading coefficient here is 12.
This leading coefficient helps in identifying the possible rational zeros of the polynomial through the Rational Zeros Theorem. The value of the leading coefficient, which is 12 in this exercise, leads us to list its positive and negative integer factors. These factors contribute to forming the denominators of potential rational zeros, significantly impacting the solutions we derive.
constant term
The constant term in a polynomial is the term without a variable; it stands alone at the end of the expression. It is an important element in polynomial functions, especially for applying the Rational Zeros Theorem:
  • In our polynomial \(U(x) = 12x^5 + 6x^3 - 2x - 8\), the constant term is \(-8\).
  • When finding potential rational zeros, the constant term gives us the list of \(p\) values, which are the potential numerators.
For instance, the factors of \(-8\) are \(\pm 1, \pm 2, \pm 4,\) and \(\pm 8\). These are combined with the factors of the leading coefficient to derive all possible rational zeros by forming fractions with these factors.
factors of integers
Factors of integers are numbers that divide the integer exactly without leaving a remainder. Understanding integer factors is vital for implementing the Rational Zeros Theorem:
  • In the exercise, we considered the factors of the constant term \(-8\): \(\pm 1, \pm 2, \pm 4, \pm 8\).
  • Similarly, we listed the factors of the leading coefficient 12: \(\pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12\).
These factors are essential for determining the potential rational zeros of the polynomial by forming all possible combinations of \(\frac{p}{q}\) where \(p\) are factors of the constant term and \(q\) are factors of the leading coefficient. This method ensures that we consider all possible rational solutions the polynomial equation might have.

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Most popular questions from this chapter

The real solutions of the given equation are rational. List all possible rational roots using the Rational Zeros Theorem, and then graph the polynomial in the given viewing rectangle to determine which values are actually solutions. (All solutions can be seen in the given viewing rectangle.) $$3 x^{3}+8 x^{2}+5 x+2=0 ; \quad[-3,3] \text { by }[-10,10]$$

A polynomial \(P\) is given. (a) Find all the real zeros of \(P\). (b) Sketch the graph of \(P\). $$P(x)=x^{5}-x^{4}-6 x^{3}+14 x^{2}-11 x+3$$

How Many Real Zeros Can a Polynomial Have? Give examples of polynomials that have the following properties, or explain why it is impossible to find such a polynomial. (a) A polynomial of degree 3 that has no real zeros (b) A polynomial of degree 4 that has no real zeros (c) A polynomial of degree 3 that has three real zeros, only one of which is rational (d) A polynomial of degree 4 that has four real zeros, none of which is rational What must be true about the degree of a polynomial with integer coefficients if it has no real zeros?

Give an example of a rational function that has vertical asymptote \(x=3 .\) Now give an example of one that has vertical asymptote \(x=3\) and horizontal asymptote \(y=2 .\) Now give an example of a rational function with vertical asymptotes \(x=1\) and \(x=-1,\) horizontal asymptote \(y=0,\) and \(x\) -intercept 4.

Find all rational zeros of the polynomial, and then find the irrational zeros, if any. Whenever appropriate, use the Rational Zeros Theorem, the Upper and Lower Bounds Theorem, Descartes' Rule of Signs, the quadratic formula, or other factoring techniques. $$P(x)=x^{5}-7 x^{4}+9 x^{3}+23 x^{2}-50 x+24$$
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