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Chapter 2: Problem 99

Find all real zeros of the function. $$f(x)=4 x^{3}-3 x-1$$

Short Answer

Expert verified
The real zeros of the function \(f(x) = 4x^{3}-3x-1\) are \(x = -1/4, 1/2, -1/2\).

Step by step solution

01

Write Down the Equation

The first step is to write down the equation that needs to be solved. In this case, that is \(4x^{3}-3x-1=0\).
02

Checking for Factorization

Cubic equations can often be factorized to make solving them easier. However, in this case, there are no simple factors of the equation \(4x^{3}-3x-1=0\).
03

Using the Rational Root Theorem

The Rational Root Theorem helps to find a first rational root by checking for factors of the constant term \(p\) and the leading coefficient \(q\), in form \(p/q\). Here it's considerably complex due to the cubic nature of the polynomial. Nonetheless, in this case, the constant term is -1 and the leading coefficient is 4, so the possible rational roots are \(\pm1, \pm1/2, \pm1/4\).
04

Evaluating the Possible Roots

Next, plug these possible roots into the equation \(4x^{3}-3x-1=0\). After testing the possible roots, it is found that only -1/4 is a root of the function.
05

Perform Synthetic Division

To find the remaining roots, perform synthetic division of the original equation by the root just found, here \(-1/4\). This simplifies the equation to a easier quadratic equation \(16x^{2} - 1 = 0\).
06

Solve the Quadratic Equation

Solving this, applying the quadratic formula, i.e., \(x = [-b \pm sqrt(b^{2}-4ac)]/(2a)\), provides the remaining two roots \(x = -1/4, 1/2, -1/2\).

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Most popular questions from this chapter

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