Question

Factor each of the following polynomials completely. Indicate any that are not factorable using integers. Don't forget to look first for a common monomial factor. $$ n^{4}-81 $$

Name: Problem 37, Chapter 3: Intermediate Algebra
Uploaded: 2022-02-27T23:09:32-08:00
Duration: 23 s
Channel: James Kiss
Description: Factor each of the following polynomials completely. Indicate any that are not factorable using integers. Don't forget to look first for a common monomial factor. n^4-81

   Factor each of the following polynomials completely. Indicate any that are not factorable using integers. Don't forget to look first for a common monomial factor.
$$
n^{4}-81
$$

Intermediate Algebra

Jerome E. Kaufmann,… 10th Edition

Chapter 3, Problem 37

Instant Answer

Solved by Expert James Kiss

02/27/2022

Step 1

The difference of squares formula is $a^2 - b^2 = (a+b)(a-b)$. Here, $a = n^2$ and $b = 9$. Show more…

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Factor each of the following polynomials completely. Indicate any that are not factorable using integers. Don't forget to look first for a common monomial factor. $$ n^{4}-81 $$

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Key Concepts

Common Monomial Factor

This concept involves checking for any common factor present in every term of the polynomial. Although factoring out a common monomial does not change the nature of the polynomial’s factorization, it simplifies the process and helps in identifying other factorization patterns easily.

Difference of Two Squares

The difference of two squares is a key factorization technique used when an expression fits the form a² - b², which factors into (a - b)(a + b). Recognizing this pattern is critical as it allows for further factorization of the polynomial into simpler binomials.

Irreducible Factor Recognition

After applying standard factorization methods, it's necessary to determine if any of the remaining factors cannot be factored further using integers. For example, certain quadratic expressions do not have integer roots, marking them as irreducible over the set of integers.

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