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Chapter 2: Problem 32

Find all real solutions of the polynomial equation. $$x^{5}-x^{4}-3 x^{3}+5 x^{2}-2 x=0$$

Short Answer

Expert verified
The real solutions of the equation are \(x=0\), \(x=1\), and \(x鈮�1.09\).

Step by step solution

01

Factor the Polynomial

Starting with the equation \(x^{5}-x^{4}-3x^{3}+5x^{2}-2x=0\), this can be factored into \(x(x^{4}-x^{3}-3x^{2}+5x-2)=0\).
02

Factor Further

This equation can be factored even further into \(x(x-1)(x^3-x^2+2x-2)=0\).
03

Solve for x

Setting each factor equal to zero gives the solutions \(x=0\), \(x=1\), and the solution from the cubic equation \(x^3-x^2+2x-2=0\) can be found by factoring or using the cubic formula. After some computations, it turns out that the cubic equation has one real root \(x鈮�1.09\) and two complex roots which are neglected as we are interested in real solutions only.
04

List All Real Solutions

Listing all real solutions gives \(x=0\), \(x=1\), and \(x鈮�1.09\).

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Most popular questions from this chapter

Use the given zero to find all the zeros of the function. Function \(f(x)=x^{4}+3 x^{3}-5 x^{2}-21 x+22\) Zero \(-3+\sqrt{2} i\)

Determine (if possible) the zeros of the function \(g\) when the function \(f\) has zeros at \(x=r_{1}, x=r_{2},\) and \(x=r_{3}\) $$g(x)=f(2 x)$$

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. $$f(x)=2 x^{4}+5 x^{3}+4 x^{2}+5 x+2$$

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

Find all real zeros of the function. $$f(z)=12 z^{3}-4 z^{2}-27 z+9$$
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