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Chapter 1: Problem 22

Find the zeros of the function algebraically. $$f(x)=4 x^{3}-24 x^{2}-x+6$$

Short Answer

Expert verified
The zeros of the function \(f(x) = 4x^{3}-24x^{2}-x+6\) are \(3, \frac{1 + \sqrt{3}}{2}\), and \(\frac{1 - \sqrt{3}}{2}\)

Step by step solution

01

Rational Roots Theorem

Use the Rational Roots Theorem to list out potential rational roots. The Rational Root theorem suggests that if a function has integer coefficients, then every rational zero will have the form \(\frac{p}{q}\) where p is a factor of the constant term (6) and q is a factor of the leading coefficient (4). For this problem, the possible rational factors of 6 are \(\pm1, \pm2, \pm3, \pm6\) and the possible rational factors of 4 are \(\pm1, \pm2, \pm4\). So, the potential rational roots include \(\pm1, \pm2, \pm3, \pm6, \pm\frac{1}{2}, \pm\frac{3}{2}\) and \(\pm\frac{3}{4}\)
02

Synthetic Division

Next, use synthetic division to determine which, if any, of the potential rational roots are actually roots of the equation. Usually, it's best to start with the smaller values. In this case, when trying -1 and -2, the synthetic division does not yield a remainder of 0. When trying \(x=3\), the synthetic division yields a remainder of 0, which indicates \(x=3\) is indeed a root. Synthetic division results also give us a simplified quadratic function: \(4x^2 - 4x - 2\).
03

Quadratic formula

To find the other zeroes of the function, the quadratic formula is used on this reduced equation. The quadratic formula is \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), yielding \(x = \frac{4 \pm \sqrt{16 + 32}}{8} = \frac{4 \pm \sqrt{48}}{8}\). These simplify to \(x = \frac{1 \pm \sqrt{3}}{2}\).
04

Provide Solution

The zeros of the cubic function therefore are \(x = 3, \frac{1 + \sqrt{3}}{2}, \frac{1 - \sqrt{3}}{2}\)

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