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Exercises Equations and inequalities

Example 5 from Algebra and Trigonometry, 2.5 Quadratic Equations

Solving a Polynomial of Higher Degree by Factoring

Solve the equation by factoring: $−3 x^{3} - 5 x^{2} - 2 x = 0.$

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

Solution
This equation does not look like a quadratic, as the highest power is 3, not 2. Recall that the first thing we want to do when solving any equation is to factor out the GCF, if one exists. And it does here. We can factor out $- x$ from all of the terms and then proceed with grouping.

$\begin{matrix} −3 x^{3} - 5 x^{2} - 2 x & = & 0 \\ - x (3 x^{2} + 5 x + 2) & = & 0 \end{matrix}$
Use grouping on the expression in parentheses.

$\begin{matrix} - x (3 x^{2} + 3 x + 2 x + 2) & = & 0 \\ - x [3 x (x + 1) + 2 (x + 1)] & = & 0 \\ - x (3 x + 2) (x + 1) & = & 0 \end{matrix}$
Now, we use the zero-product property. Notice that we have three factors.

$\begin{matrix} - x & = & 0 \\ x & = & 0 \\ 3 x + 2 & = & 0 \\ x & = & - \frac{2}{3} \\ x + 1 & = & 0 \\ x & = & −1 \end{matrix}$

The solutions are $0,$ $- \frac{2}{3},$ and $−1.$

Your turn

The book's Try It right after this example: same idea, new numbers. Only the answer is given.

Solve by factoring: $x^{3} + 11 x^{2} + 10 x = 0.$

How did it go?

PreviousSolving a Quadratic Equation Using Grouping Next Solving a Simple Quadratic Equation Using the Square Root Property

From Algebra and Trigonometry (1st edition) by OpenStax (Rice University), section 2.5, licensed under CC BY 4.0. Original at openstax.org. We split the solution into steps; the words, numbers and figures are the book's. OpenStax doesn't endorse this site.