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Chapter 3: Problem 44

Solve the equation \(2 x^{3}-3 x^{2}-11 x+6=0\) given that \(-2\) is a zero of \(f(x)=2 x^{3}-3 x^{2}-11 x+6\)

Short Answer

Expert verified
The roots of the given equation are -2, and the roots of the quadratic equation obtained from the division.

Step by step solution

01

Confirm the given root

We have been given that -2 is a root of the polynomial. We can check this by substituting -2 for x in the equation: \(f(-2) = 2*(-2)^3 - 3*(-2)^2 - 11*(-2) + 6\). If the result of this computation is zero, then -2 is indeed a root.
02

Divide the Polynomial

We can find the other roots by performing polynomial division. Here, we will divide the polynomial \(2x^3 - 3x^2 - 11x + 6\) by the binomial \(x - (-2)\), which is \(x+2\). The quotient we get is another, simpler polynomial whose roots are also the roots of our original polynomial.
03

Solving the Quotient Polynomial

Now, we solve the polynomial resulting from the division in step 2. This polynomial will be a quadratic equation which can be solved by factoring, completing the square, or using the quadratic formula.

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Most popular questions from this chapter

a. Use a graphing utility to graph \(y-2 x^{2}-82 x+720\) in a standard viewing rectangle. What do you observe? b. Find the coordinates of the vertex for the given quadratic function. c. The answer to part (b) is \((20.5,-120.5) .\) Because the leading coefficient, \(2,\) of the given function is positive, the vertex is a minimum point on the graph. Use this fact to help find a viewing rectangle that will give a relatively complete picture of the parabola. With an axis of symmetry at \(x=20.5,\) the setting for \(x\) should extend past this, so try \(\mathrm{Xmin}=0\) and \(\mathrm{Xmax}=30 .\) The setting for \(y\) should include (and probably go below) the \(y\) -coordinate of the graph's minimum y-value, so try Ymin \(=-130\). Experiment with Ymax until your utility shows the parabola's major features. d. In general, explain how knowing the coordinates of a parabola's vertex can help determine a reasonable viewing rectangle on a graphing utility for obtaining a complete picture of the parabola.

Write the equation of each parabola in standard form. Vertex: \((-3,-4) ;\) The graph passes through the point \((1,4)\)

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. $$\frac{5 x^{2}}{x^{2}-4} \cdot \frac{x^{2}+4 x+4}{10 x^{3}}$$

Find the vertex for each parabola. Then determine a reasonable viewing rectangle on your graphing utility and use it to graph the quadratic function. $$y= 5 x^{2}+40 x+600$$

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