/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 45 Solve the equation \(12 x^{3}+16... [FREE SOLUTION] | 91影视

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Chapter 3: Problem 45

Solve the equation \(12 x^{3}+16 x^{2}-5 x-3=0\) given that \(-\frac{3}{2}\) is a root.

Short Answer

Expert verified
The roots of the equation are \(x = -\frac{3}{2}\), \(x = \frac{1 + \sqrt{7}}{6}\), and \(x = \frac{1 - \sqrt{7}}{6}\).

Step by step solution

01

Verify Given Root

Substitute the given root -3/2 into the equation to verify that it is indeed a root: \(12(-\frac{3}{2})^{3}+16(-\frac{3}{2})^{2}-5(-\frac{3}{2})-3 = 0\). Since this simplifies to zero, it confirms that -3/2 is a root.
02

Carry Out Polynomial Division

Next we need to divide the original polynomial by the binomial corresponding to the given root to get a quadratic polynomial. The binomial is \(x-(-\frac{3}{2}) = x + \frac{3}{2}\). The process of polynomial division yields a quadratic equation \(12x^{2} - 2x -2 =0\).
03

Solve the Quadratic Equation

Next, we solve the quadratic equation \(12x^{2} - 2x -2 =0\) for its roots. By using the quadratic formula \(x=\frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}\), we find that the roots are \(x = \frac{1 \pm \sqrt{7}}{6}\).

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Most popular questions from this chapter

Make Sense? In Exercises \(94-97\), determine whether each statement makes sense or does not make sense, and explain your reasoning. When solving \(f(x)>0,\) where \(f\) is a polynomial function, 1 only pay attention to the sign of \(f\) at each test value and not the actual function value.

What is a rational inequality?

The equation for \(f\) is given by the simplified expression that results after performing the indicated operation. Write the equation for \(f\) and then graph the function. $$\frac{2}{x^{2}+3 x+2}-\frac{4}{x^{2}+4 x+3}$$

Is every rational function a polynomial function? Why or why not? Does a true statement result if the two adjectives rational and polynomial are reversed? Explain.

Follow the seven steps on page 399 to graph each rational function. $$f(x)=\frac{x-2}{x^{2}-4}$$
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