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 the real roots of a cubic equation, particularly by using a graphing utility (such as a graphing calculator or graphing software) to approximate the solutions.

Key Terms and Formulas

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

• Real Solution: A value of $x$ that makes the equation true and is a real number.

• Graphing Utility: A tool (calculator or software) used to plot the function and visually identify where it crosses the $x$-axis (the real roots).

Step-by-Step Guidance

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

2. Enter the function $f(x) = x^3 - 8x + 1$ into your graphing utility.

3. Graph the function and observe where the curve crosses the $x$-axis. These $x$-values are the real solutions to the equation.

4. Use the graphing utility's 'zero' or 'root' feature to approximate the $x$-values where $f(x) = 0$. Record these values to the desired decimal places.

Try solving on your own before revealing the answer!

Final Answer:

The real solutions to $x^3 - 8x + 1 = 0$ (approximated using a graphing utility) are:

$x \approx -2.879$, $x \approx -0.125$, and $x \approx 3.004$ (rounded to three decimal places).

These are the $x$-values where the graph of $f(x)$ crosses the $x$-axis.

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

Background

Topic: Solving Polynomial Equations

This question is about solving a polynomial equation that involves both cubic and quadratic terms. The goal is to rearrange the equation so that all terms are on one side, then use algebraic or graphical methods to find the solutions.

Key Terms and Formulas

• Polynomial Equation: An equation involving terms with variables raised to whole number powers.

• Standard Form: All terms on one side set equal to zero: $ax^3 + bx^2 + cx + d = 0$.

Step-by-Step Guidance

1. Start by moving 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. Rearrange the terms in descending order of degree: $-x^3 - \frac{4}{3}x^2 + \frac{5}{2}x + 2 = 0$.

3. If desired, multiply both sides by a common denominator (such as 6) to clear the fractions and make the equation easier to work with.

4. Once the equation is in standard form with integer coefficients, use a graphing utility or algebraic methods to approximate or solve for the real roots.

Try solving on your own before revealing the answer!

Final Answer:

After clearing fractions and rearranging, the equation becomes:

$-6x^3 - 8x^2 + 15x + 12 = 0$

Using a graphing utility, the approximate real solutions are:

$x \approx -2.000$, $x \approx -0.667$, and $x \approx 1.500$ (rounded to three decimal places).

These are the $x$-values where the polynomial equals zero.