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Chapter 3: Problem 77

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 rational roots of the polynomial function are \{-\frac{3}{2}, \frac{3}{2}, -2, 2\}.

Step by step solution

01

Putting the Polynomial expression in quadratic form

By changing the variable, let's say \(y = x^2\), we can express the given polynomial in the quadratic form as follows: \(4y^2 - 25y + 36 = 0\).
02

Factoring the quadratic form

Now the quadratic equation can be factored: \(4y^2 - 25y + 36 = 0\) which will give us \((4y - 9)(y - 4) = 0\)
03

Finding roots for y

Without fail, y should equate to 0, this will yield two possible values for y: \(y1 = \frac{9}{4}\) and \(y2 = 4\). Since y = x^2, then x = 卤鈭歽.
04

Finding values for x

For each value of y, we calculate the related values for x. For \(y1 = \frac{9}{4}\) we will have \(x1 = 卤\frac{3}{2}\) and for \(y2 = 4\) we will have \(x2 = 卤2\). Hence, the rational roots of the polynomial function \(P(x) = x^4 - \frac{25}{4}x^2 + 9\) are \(\{-\frac{3}{2}, \frac{3}{2}, -2, 2\}\)

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Most popular questions from this chapter

(a) use Descartes's Rule of Signs to determine the possible numbers of positive and negative real zeros of \(f\) (b) list the possible rational zeros of \(f,\) (c) use a graphing utility to graph \(f\) so that some of the possible zeros in parts (a) and (b) can be disregarded, and (d) determine all the real zeros of \(f\). $$f(x)=x^{4}-x^{3}-29 x^{2}-x-30$$

Use a graphing utility to graph the function. Explain why there is no vertical asymptote when a superficial examination of the function might indicate that there should be one. $$h(x)=\frac{6-2 x}{3-x}$$

Use synthetic division to verify the upper and lower bounds of the real zeros of \(f .\) Then find all real zeros of the function. \(f(x)=x^{4}-4 x^{3}+15\) Upper bound: \(x=4\) Lower bound: \(x=-1\)

Determine whether the statement is true or false. Justify your answer. If \((7 x+4)\) is a factor of some polynomial function \(f\) then \(\frac{4}{7}\) is a zero of \(f\).

Divide using long division. $$\left(4 x^{5}+3 x^{3}-10\right) \div(2 x+3)$$
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