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Chapter 2: Problem 103

Find the rational zeros of the polynomial function. $$P(x)=x^{4}-\frac{25}{4} x^{2}+9=\frac{1}{4}\left(4 x^{4}-25 x^{2}+36\right)$$

Short Answer

Expert verified
The process of factoring the equation uncovers the rational zeros of the polynomial. The value for x obtained in the final step is the solution.

Step by step solution

01

Rewrite the Function

Transform the polynomial function to a form that resembles a quadratic function by identifying the mid-term for further simplification. The equation becomes \((2x^2)^2 - (\frac{5}{2})2x^2 + 3^2\)
02

Apply the Formula for a Quadratic Equation

We can now identify \(a\), \(b\), and \(c\) in the standard quadratic form \(az^2+bz+c\), where \(z = 2x^2\). We get \(a = 1\), \(b = -\frac{5}{2}\), and \(c = 3^2\). The roots of this quadratic equation in \(z\) are found using the quadratic formula \( z = [-b \pm \sqrt{b^2-4ac}] / (2a)\): \[z = [\frac{5}{2} \pm \sqrt{(-\frac{5}{2})^2-4*1*3^2}] / (2*1)\]
03

Find the value of z

Simplify to get z. Here, \(z = 2x^2\). Therefore, after finding the values of \(z\), we solve for \(x\). Note that we are only interested in the rational solutions.
04

Solve for x

Now that we have the values of \(z\), solving for \(x\) will give us the rational zeros of the polynomial. Consider both the square root case and the negative square root case.

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Most popular questions from this chapter

Write the polynomial as the product of linear factors and list all the zeros of the function. $$f(x)=x^{4}-16$$

Match the cubic function with the numbers of rational and irrational zeros. (a) Rational zeros: \(0 ;\) irrational zeros: 1 (b) Rational zeros: \(3 ;\) irrational zeros: 0 (c) Rational zeros: \(1 ;\) irrational zeros: 2 (d) Rational zeros: \(1 ;\) irrational zeros: 0 $$f(x)=x^{3}-1$$

The coordinate system shown below is called the complex plane. In the complex plane, the point that corresponds to the complex number \(a+b i\) is \((a, b)\) (GRAPH CANNOT COPY) Match each complex number with its corresponding point. (i) 3 (ii) \(3 i\) (iii) \(4+2 i\) (iv) \(2-2 i\) (v) \(-3+3 i\) (vi) \(-1-4 i\)

Simplify the complex number and write it in standard form. $$(-i)^{6}$$

Find all the zeros of the function. When there is an extended list of possible rational zeros, use a graphing utility to graph the function in order to disregard any of the possible rational zeros that are obviously not zeros of the function. $$g(x)=x^{5}-8 x^{4}+28 x^{3}-56 x^{2}+64 x-32$$
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