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Chapter 0: Problem 78

Factor completely, or state that the polynomial is prime. $$x^{2}+36$$

Short Answer

Expert verified
The polynomial \(x^{2} + 36\) is prime.

Step by step solution

01

Identify the Type of Expression

This is a binomial, an expression with two terms. It is of the form \(x^2 + a^2\), which is a sum of squares.
02

Determine If It Can Be Factored

A sum of squares, such as our expression \(x^2 + 36\) or more generally \(x^2 + a^2\), cannot be factored over the set of real numbers. In the complex numbers, it can be factored as \((x + ai)(x - ai)\), where 'i' is the imaginary unit. However, if we stick to real numbers, it can't be factored and the polynomial is a prime polynomial.

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Most popular questions from this chapter

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The equation \(a x^{2}+c=0, a \neq 0,\) cannot be solved by the quadratic formula.

Use the strategy for solving word problems, modeling the verbal conditions of the problem with a linear inequality. An elevator at a construction site has a maximum capacity of 3000 pounds. If the elevator operator weighs 245 pounds and each cement bag weighs 95 pounds, how many bags of cement can be safely lifted on the elevator in one trip?

When graphing the solutions of an inequality, what does a parenthesis signify? What does a square bracket signify?

In solving \(\sqrt{2 x-1}+2=x,\) why is it a good idea to isolate the radical term? What if we don't do this and simply square each side? Describe what happens.

$$\text { Solve for } t: \quad s=-16 t^{2}+v_{0} t$$
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