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Chapter 2: Problem 41

Find the zeros of each polynomial function. If a zero is a multiple zero, state its multiplicity. $$P(x)=6 x^{4}+23 x^{3}+19 x^{2}-8 x-4$$

Short Answer

Expert verified
Without complete factoring of this polynomial, the particular zeros and their multiplicities cannot be stated. Factoring this high degree polynomial might require other techniques not shown in this solution, such as using a quadratic formula or using a graphing calculator to estimate real and complex roots.

Step by step solution

01

Set the Polynomial Equal to Zero

To find the possible zeros or roots of the polynomial, it should first be set equal to zero. Thus, the equation to solve is: \[6x^{4}+23x^{3}+19x^{2}-8x-4=0\],
02

Apply Rational Root Theorem

If the polynomial is difficult to factorize by simple inspection, the Rational Root Theorem can be used. The Rational Root Theorem states that any rational root, expressed in lowest terms, has a numerator that is a factor of the constant term, and a denominator that is a factor of the leading coefficient. For this polynomial, potential rational roots can be \(卤1, 卤2, 卤4\) (factors of -4) divided by \(卤1, 卤2, 卤3, 卤6\) (factors of 6). Test these to see which of these are possible roots by either synthetic division or substitution into the original equation.
03

Factorize the Polynomial

If any rational roots were found in Step 2, use synthetic division or the Factor Theorem to find the remaining polynomial, then continue to factorize as far as possible. If no rational roots are found, other methods (graphing, use of quadratic formula, etc) may be needed.
04

State the Multiplicity

Once the polynomial is completely factorized, the multiplicity of each root is given by the powers of the factors in the factorization. If a root only appears once, it is of multiplicity one. If it appears twice (as a square of a factor), it is of multiplicity two, and so on.

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Most popular questions from this chapter

In Exercises 61 to 70 , use the quadratic formula to solve each quadratic equation. $$8 x^{2}+12 x=-17$$

DIGITS OF PI In 1999, Professor Yasumasa Kanada of the University of Tokyo used a supercomputer to compute 206,158,430,000 digits of pi ( \(\pi\) ). (Source: Guinness World Records 2001, Bantam Books, p. \(252 .\) ) Computer scientists often try to find mathematical models that approximate the time a computer program takes to complete a calculation or mathematical procedure. Procedures for which the completion time can be closely modeled by a polynomial are called polynomial time procedures. Here is an example. A student finds that the time, in seconds, required to compute \(n \times 10,000\) digits of pi on a personal computer using the mathematical program MAPLE is closely approximated by $$\begin{aligned} T(n)=& 0.23245 n^{3}+0.53797 n^{2} \\ &+7.88932 n-8.53299 \end{aligned}$$ a. Evaluate \(T(n)\) to estimate how long, to the nearest second, the computer takes to compute 50,000 digits of \(\mathrm{pi}\). b. About how many digits of pi can the computer compute in 5 minutes? Round to the nearest thousand digits.

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