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Exercises Polynomials and factoring

Example 2 from Algebra and Trigonometry, 5.5 Zeros of Polynomial Functions

Using the Factor Theorem to Find the Zeros of a Polynomial Expression

Show that $(x + 2)$ is a factor of $x^{3} - 6 x^{2} - x + 30.$ Find the remaining factors. Use the factors to determine the zeros of the polynomial.

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
We can use synthetic division to show that $x^{3} - 6 x^{2} - x + 30$ is a factor of the polynomial.

The remainder is zero, so $(x + 2)$ is a factor of the polynomial. We can use the Division Algorithm to write the polynomial as the product of the divisor and the quotient:

$(x + 2) (x^{2} - 8 x + 15)$
We can factor the quadratic factor to write the polynomial as

$(x + 2) (x - 3) (x - 5)$

By the Factor Theorem, the zeros of $x^{3} - 6 x^{2} - x + 30$ are –2, 3, and 5.

Your turn

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

Use the Factor Theorem to find the zeros of $f (x) = x^{3} + 4 x^{2} - 4 x - 16$ given that $(x - 2)$ is a factor of the polynomial.

How did it go?

PreviousUsing the Remainder Theorem to Evaluate a Polynomial Next Listing All Possible Rational Zeros

From Algebra and Trigonometry (1st edition) by OpenStax (Rice University), section 5.5, 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.