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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 Form

Looking at the polynomial \(x^{2}+64\), one can see that it takes the form of \(a^{2}+b^{2}\) where \(a=x\) and \(b=8\), since \(8^{2} = 64\).
02

Recognize Prime Polynomials

Over the set of real numbers, a sum of squares such as \(a^{2}+b^{2}\) can't be factored. It is therefore defined as a prime polynomial.
03

Statement

Since it cannot be factored, we simply state that the polynomial \(x^{2}+64\) is prime.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Factoring Polynomials
Factoring polynomials is an essential part of algebra that involves breaking down a polynomial into simpler "building blocks," called factors, that when multiplied together reproduce the original polynomial. This concept is similar to factoring numbers into prime components.
For example, the number 12 can be factored into 2 \( \times \) 2 \( \times \) 3, which are all prime numbers. With polynomials, the goal is the same: simplify the expression to reveal its simpler parts.
  • Factoring uses different techniques, such as identifying common factors, using special polynomial identities, and applying algebraic strategies.
  • A completely factored polynomial has no factors that can be further divided except by 1 or -1.
  • Polynomials are sometimes "prime," meaning they cannot be factored further within the set of real numbers.
When you try to factor a sum of squares, you'll often find that it's already in its simplest form under real numbers, making it prime.
Exploring the Sum of Squares
The sum of squares refers to the expression of two squared terms added together, written as \(a^{2} + b^{2}\). Unlike the "difference of squares," where \(a^{2} - b^{2}\) can be factored, the sum of squares is not factorizable when dealing with real numbers.
This is an essential point to remember:
  • In real number arithmetic, formulas and identities we use to factor expressions like \(a^{2} - b^{2}\) do not extend to a simple equivalent for \(a^{2} + b^{2}\).
  • The sum of squares cannot be transformed into a product of binomials without introducing complex numbers.
You might encounter complex factorizations for sum of squares in more advanced math, but within the scope of real numbers, what's remarkable is that such expressions remain prime.
The Role of Real Numbers
In mathematics, real numbers include all the numbers that can be found on the number line. This includes rational numbers such as integers and fractions, as well as irrational numbers that cannot be expressed as simple fractions, like \(\pi\) or \(\sqrt{2}\).
Real numbers are fundamentally important for understanding algebra and calculus because they form the basis of most calculations.
  • When factoring polynomials using real numbers, some expressions remain "prime," meaning they cannot be factored further within the real number system.
  • This limitation is because certain mathematical transformations, like factoring the sum of squares, require stepping outside real numbers into the domain of complex numbers.
Understanding the scope of real numbers helps clarify why some polynomials resist factorization when constrained by this set, emphasizing the boundaries set by the real number system.

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Most popular questions from this chapter

Exercises \(159-161\) will help you prepare for the material covered in the next section. In parts (a) and (b), complete each statement. a. \(\frac{b^{7}}{b^{3}}=\frac{b \cdot b \cdot b \cdot b \cdot b \cdot b \cdot b}{b \cdot b \cdot b}=b^{2}\) b. \(\frac{b^{8}}{b^{2}}=\frac{b \cdot b \cdot b \cdot b \cdot b \cdot b \cdot b \cdot b}{b \cdot b}=b^{?}\) c. Generalizing from parts (a) and (b), what should be done with the exponents when dividing exponential expressions with the same base?

Will help you prepare for the material covered in the first section of the next chapter. If \(y=4-x^{2},\) find the value of \(y\) that corresponds to values of \(x\) for each integer starting with \(-3\) and ending with 3

Factor completely. $$x^{2 n}+6 x^{n}+8$$

Determine whether each statement makes sense or does not make sense, and explain your reasoning. The model \(P=-0.18 n+2.1\) describes the number of pay phones, \(P,\) in millions, \(n\) years after \(2000,\) so I have to solve a linear equation to determine the number of pay phones in 2010

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