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Chapter 4: Problem 35

Find all rational zeros of the polynomial, and write the polynomial in factored form. $$ P(x)=4 x^{3}-7 x+3 $$

Short Answer

Expert verified
The rational zeros are 1 and -3; the factored form is \( (x - 1)(x + 3)(4x + 3) \).

Step by step solution

01

Identify Possible Rational Roots

To find possible rational roots, use the Rational Root Theorem. This theorem suggests that the possible rational roots are the factors of the constant term divided by the factors of the leading coefficient. The constant term is 3, and the leading coefficient is 4. The factors of 3 are 卤1, 卤3, and the factors of 4 are 卤1, 卤2, 卤4. Therefore, the possible rational roots are 卤1, 卤3, 卤1/2, 卤3/2, 卤1/4, and 卤3/4.
02

Test Possible Rational Roots Using Synthetic Division

Use synthetic division to test each possible rational root until one is found that results in a remainder of zero. Start testing with x = 1. Upon using synthetic division, the remainder is not zero. Continue with x = -1, 3, and -3, until finding a root that gives a remainder of zero.
03

Verify and Factor Polynomial

After testing, you find that x = 1 and x = -3 are roots. Using synthetic division with these roots confirms they yield a remainder of zero. Therefore, the polynomial can be divided by (x - 1)(x + 3), resulting in a quotient of 4x + 3, so the polynomial is factorable as (x - 1)(x + 3)(4x + 3).
04

Write Polynomial in Factored Form

Now that you have identified the rational roots and verified them, express the polynomial in factored form as: \[ P(x) = (x - 1)(x + 3)(4x + 3) \].

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Key Concepts

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

Synthetic Division
Synthetic division is a simplified method to divide a polynomial by a binomial of the form \(x - r\). It is similar to long division but requires less writing and is quicker to perform. To start, list the coefficients of the polynomial in a row. Then, write the root being tested outside this row on the left.
Follow these steps:
  • Bring down the leading coefficient directly below the line.
  • Multiply this number by the root you've written outside and place the result under the next coefficient.
  • Add the numbers in this column, write the result below, and repeat the process until completion.
  • If the final number (remainder) is zero, the tested root is indeed a root of the polynomial.
This method helps identify one possible root at a time from the list generated using the Rational Root Theorem. Once discovered, it can confirm rational roots very efficiently and helps in factoring the polynomial step-by-step.
Factoring Polynomials
Factoring polynomials involves expressing the polynomial as a product of its factors. Once potential rational roots are determined using the Rational Root Theorem and verified via synthetic division, the next step is factoring.
For example, if the roots of a polynomial are \(1\) and \(-3\), as found using synthetic division, these roots correspond to the factors \((x - 1)\) and \((x + 3)\). The remaining quotient from synthetic division, which is a lower degree polynomial, is also a factor of the original polynomial.
The key steps for factoring:
  • Identify all rational roots using synthetic division.
  • Write each root as a binomial factor \((x - r)\).
  • Multiply these factors with any remaining unfactored polynomial quotient to get the fully factored form.
Once in factored form, the polynomial simplifies analysis, making calculations like solving for zeros much more straightforward.
Polynomial Zeros
The zeros of a polynomial are the values of \(x\) that make the polynomial equal to zero. In other words, they are the solutions to the equation \(P(x) = 0\). In the context of the Rational Root Theorem, these zeros can often be rational numbers that also serve as factors of the polynomial.
Here's how to find polynomial zeros:
  • Use the Rational Root Theorem to find potential rational zeros.
  • Test these potential zeros via synthetic division to confirm which ones are actual zeros.
  • Every confirmed root is a zero, and for every zero found, the polynomial degree decreases, revealing more potential zeros.
Finding all zeros of a polynomial gives you insight into its behavior and characteristics. Once all zeros are identified, you can express the polynomial as a product of its linear factors. This enables further exploration of the polynomial's properties, such as graphing or solving related problems involving polynomial equations.

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Most popular questions from this chapter

Show that the given values for \(a\) and \(b\) are lower and upper bounds for the real zeros of the polynomial. $$ P(x)=3 x^{4}-17 x^{3}+24 x^{2}-9 x+1 ; \quad a=0, b=6 $$

The Doppler Effect As a train moves toward an observer (see the figure), the pitch of its whistle sounds higher to the ob- server than it would if the train were at rest, because the crests of the sound waves are compressed closer together. This phenomenon is called the Doppler effect. The observed pitch \(P\) is a function of the speed \(v\) of the train and is given by $$ P(v)=P_{0}\left(\frac{s_{0}}{s_{0}-v}\right) $$ where \(P_{0}\) is the actual pitch of the whistle at the source and \(S_{0}=332 \mathrm{m} / \mathrm{s}\) is the speed of sound in air. Suppose that a train has a whistle pitched at \(P_{0}=440 \mathrm{Hz}\) . Graph the func- tion \(y=P(v)\) using a graphing device. How can the vertical asymptote of this function be interpreted physically?

Find all the real zeros of the polynomial. Use the quadratic formula if necessary, as in Example \(3(\mathrm{a}) .\) $$ P(x)=x^{4}+2 x^{3}-2 x^{2}-3 x+2 $$

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' Zule of Signs, the Uuadratic formula, or other factoring techniques. $$ P(x)=2 x^{4}+3 x^{3}-4 x^{2}-3 x+2 $$

Find the slant asymptote, the vertical asymptotes, and sketch a graph of the function. $$ r(x)=\frac{x^{2}-2 x-8}{x} $$
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