Solving Cubic Equations and Polynomial Equations in College Algebra

Q10. Use a graphing utility to approximate the real solutions, if any, to the equation: $x^3 - 8x + 1 = 0$

Background

Topic: Solving Cubic Equations

This question tests your ability to find real solutions to a cubic equation using graphing technology or algebraic methods. Cubic equations can have one or three real roots, and graphing helps visualize where the function crosses the x-axis.

Key Terms and Formulas:

• Cubic Equation: An equation of the form $ax^3 + bx^2 + cx + d = 0$.

• Real Solution: A value of $x$ where the equation equals zero and $x$ is a real number.

• Graphing Utility: A calculator or software that can plot the function and help estimate roots.

Step-by-Step Guidance

1. Rewrite the equation in standard form: $x^3 - 8x + 1 = 0$.

2. Set $y = x^3 - 8x + 1$ and consider the graph of $y$ versus $x$.

3. Use a graphing utility to plot the function. Look for points where the graph crosses the x-axis; these are the real solutions.

4. Estimate the x-values where $y = 0$ by zooming in on the graph or using the root-finding feature of your graphing tool.

Try solving on your own before revealing the answer!

Final Answer:

The real solutions to $x^3 - 8x + 1 = 0$ are approximately $x \approx -2.879$, $x \approx -0.127$, and $x \approx 3.006$ (rounded to three decimal places).

These values were found by graphing the function and identifying where it crosses the x-axis.

Q11. Solve the equation: $2 - \frac{4}{3}x^2 = x^3 - \frac{5}{2}x$

Background

Topic: Solving Polynomial Equations

This question tests your ability to solve a polynomial equation involving cubic and quadratic terms. You may need to rearrange the equation and use algebraic or graphing methods to find the real solutions.

Key Terms and Formulas:

• Polynomial Equation: An equation involving terms with powers of $x$.

• Standard Form: All terms moved to one side, set equal to zero.

• Graphing Utility: Useful for approximating real roots.

Step-by-Step Guidance

1. Move all terms to one side to set the equation equal to zero: $2 - \frac{4}{3}x^2 - x^3 + \frac{5}{2}x = 0$.

2. Rewrite the equation in standard form: $-x^3 - \frac{4}{3}x^2 + \frac{5}{2}x + 2 = 0$.

3. Set $y = -x^3 - \frac{4}{3}x^2 + \frac{5}{2}x + 2$ and plot this function using a graphing utility.

4. Look for points where the graph crosses the x-axis to estimate the real solutions.

Try solving on your own before revealing the answer!

Final Answer:

The real solutions to $2 - \frac{4}{3}x^2 = x^3 - \frac{5}{2}x$ are approximately $x \approx -2.073$, $x \approx -0.468$, and $x \approx 2.541$ (rounded to three decimal places).

These values were found by graphing the function and identifying where it crosses the x-axis.