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Chapter 0: Problem 77

Factor completely, or state that the polynomial is prime. $$x^{2}+64$$

Short Answer

Expert verified
The polynomial \(x^{2}+64\) is prime.

Step by step solution

01

Identify the polynomial

Given the polynomial \(x^{2}+64\).
02

Testing for factorability

To check if a polynomial can be factored, try using common factoring techniques such as finding common factors, grouping, or testing if it is a perfect square trinomial. In this case, no common factors exist and it is not a perfect square trinomial. The polynomial is a binomial. Notice if the binomial is a difference of squares, or a sum of squares in this case.
03

Factoring the polynomial

In algebra, a sum of squares such as \(x^{2}+64\) cannot be factored over the set of real numbers. Therefore, the polynomial \(x^{2}+64\) is prime because it cannot be factored.

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Most popular questions from this chapter

Exercises \(142-144\) will help you prepare for the material covered in the next section. Use the distributive property to multiply: $$ 2 x^{4}\left(8 x^{4}+3 x\right) $$

Explain how to divide rational expressions.

In Exercises \(133-136\), determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$\frac{\sqrt{20}}{8}=\frac{\sqrt{10}}{4}$$

Rationalize the numerator. $$\frac{\sqrt{x}+\sqrt{y}}{x^{2}-y^{2}}$$

Explain how to add rational expressions having no common factors in their denominators. Use \(\frac{3}{x+5}+\frac{7}{x+2}\) in your explanation.
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