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Chapter 4: Problem 64

For each polynomial function: A. Find the rational zeros and then the other zeros; that is, solve \(f(x)=0\) B. Factor \(f(x)\) into linear factors. $$f(x)=3 x^{4}-4 x^{3}+x^{2}+6 x-2$$

Short Answer

Expert verified
Rational zeros are \(x = 1\) and \(x = -1\). Factored form: \(f(x) = 3(x + 1)(x - 1)(x - \frac{2 + i \sqrt{2}}{3})(x - \frac{2 - i \sqrt{2}}{3})\).

Step by step solution

01

Identify possible rational zeros

Use the Rational Root Theorem, which states that any rational solution of the polynomial equation must be a fraction \(\frac{p}{q}\), where p is a factor of the constant term (-2) and q is a factor of the leading coefficient (3). Possible rational zeros are: \(\pm 1, \pm 2, \pm \frac{1}{3}, \pm \frac{2}{3}\).
02

Test possible rational zeros

Use synthetic division to test each possible rational zero until we find one that gives a remainder of 0. First test \(x = 1\). The synthetic division of 3, -4, 1, 6, -2 by 1 results in a remainder, so try another value. Test \(x = -1\). The synthetic division of 3, -4, 1, 6, -2 by -1 results in zero remainder: \(3x^3 - 7x^2 + 8x - 2\).
03

Find additional zeros

Use the depressed polynomial \(3x^3 - 7x^2 + 8x - 2\) obtained from Step 2. Repeat Rational Root Theorem and test \(x = 1\). The synthetic division shows that \(x = 1\) is also a zero. The polynomial is now further reduced: \(3x^2 - 4x + 2\).
04

Solve quadratic equation

Now use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) to solve \(3x^2 - 4x + 2 = 0\). Compute the discriminant: \[b = -4, a = 3, c = 2 \implies (-4)^2 - 4(3)(2) = 16 - 24 = -8\]. Therefore, no real zeros are found. The complex zeros are \( \frac{2 \pm i \sqrt{2}}{3} \).
05

Factor polynomial

Combine found zeros and factorize the polynomial. The zeros are \(x = 1, x = -1\), and \(\frac{2 \pm i \sqrt{2}}{3}\). Thus, the polynomial can be factored as: \[ f(x) = 3(x + 1)(x - 1)(x - \frac{2 + i \sqrt{2}}{3})(x - \frac{2 - i \sqrt{2}}{3})\].

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Key Concepts

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

Rational Root Theorem
To find the rational zeros of a polynomial, we use the Rational Root Theorem.
This theorem helps us identify possible rational solutions (zeros) of a polynomial equation.
It states any possible rational zero can be expressed as \(\frac{p}{q}\).
Here,
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Most popular questions from this chapter

List the critical values of the related function. Then solve the inequality. $$\frac{4}{x^{2}-9} < \frac{3}{x^{2}-25}$$

Use synthetic division to divide. $$\left(x^{3}+3 i x^{2}-4 i x-2\right) \div(x+i)$$

The population \(P,\) in thousands, of a resort community is given by $$ P(t)=\frac{500 t}{2 t^{2}+9} $$ where \(t\) is the time, in months, since the city council raised the property taxes. (Graph can't copy) a) Find the population at \(t=0,1,3,\) and 8 months. b) Find the horizontal asymptote of the graph and complete the following: $$P(t) \rightarrow \text{ _____ } as \quad t \rightarrow \infty$$ c) Explain the meaning of the answer to part (b) in terms of the application.

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Find a rational function that satisfies the given conditions. Answers may vary, but try to give the simplest answer possible. Vertical asymptotes \(x=-4, x=5 ;\) horizontal asymptote \(y=\frac{3}{2} ; x\) -intercept \((-2,0)\)
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