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Chapter 2: 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 given cubic equation are \(x = -\frac{3}{2}\), \(x = \text{solution of simplified quadratic equation}\)

Step by step solution

01

Polynomial Division

We use the rule that if \(r\) is a root of \(p(x)\), then \(p(x)\) is divisible by \(x - r\). Given that \(-\frac{3}{2}\) is a root of the polynomial, we can divide the entire polynomial by \(x - (-\frac{3}{2})\), which is \(x + \frac{3}{2}\). This gives us a quadratic equation.
02

Simplify the Quadratic Equation

Now, we simplify the quadratic equation that we got from the polynomial division. After simplification, the quadratic equation will be in the standard form of \(ax^{2} + bx + c = 0\)
03

Solve the Quadratic Equation

Now, we use the quadratic formula, which is \(x = \frac{-b 卤 \sqrt{b^{2} - 4ac}}{2a}\) to find the roots of the quadratic equation. This will give us two additional roots of the original cubic equation.

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Most popular questions from this chapter

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The inequality \(\frac{x-2}{x+3}<2\) can be solved by multiplying both sides by \((x+3)^{2}, x \neq-3,\) resulting in the equivalent inequality \((x-2)(x+3)<2(x+3)^{2}\).

Find the horizontal asymptote, if there is one, of the graph of rational function. $$f(x)=\frac{-3 x+7}{5 x-2}$$

a. If \(y=k x^{2},\) find the value of \(k\) using \(x=2\) and \(y=64\). b. Substitute the value for \(k\) into \(y=k x^{2}\) and write the resulting equation. c. Use the equation from part (b) to find \(y\) when \(x=5\).

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. $$\frac{1-\frac{3}{x+2}}{1+\frac{1}{x-2}}$$

If \(S=\frac{k A}{P},\) find the value of \(k\) using \(A=60,000, P=40,\) and \(S=12,000\).
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