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Chapter 6: Problem 45

Factor. If a polynomial can't be factored, write "prime." $$ a^{2}+b^{2} $$

Short Answer

Expert verified
The polynomial is prime as it cannot be factored further over the real numbers.

Step by step solution

01

Understand the Expression

We have the polynomial \( a^2 + b^2 \). It is a sum of two squares where both \( a \eq 0 \) and \( b \eq 0 \). Typically, sums of squares do not factor over the real numbers.
02

Recall Factoring Rules

Generally, sums of squares like \( a^2 + b^2 \) cannot be factored into simpler polynomials with real coefficients. There is no real-number factoring similar to the difference of squares \( a^2 - b^2 = (a - b)(a + b) \).
03

Conclude the Factoring

Since \( a^2 + b^2 \) does not have any real factors, the polynomial does not factor over the real numbers. Therefore, it is considered a prime polynomial in the realm of real numbers.

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Key Concepts

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

Sum of Squares
The sum of squares is a term used in mathematics to describe an algebraic expression that combines two squared variables with a 'plus' operator. A classic example is the expression \( a^2 + b^2 \). Unlike 'difference of squares', which can be factored easily into real numbers
  • \( a^2 - b^2 = (a-b)(a+b) \)
a sum of squares generally doesn't factorize over the real numbers. This means expressions like \( a^2 + b^2 \) cannot be broken down into a product of two simpler polynomials with real coefficients. In a sum of squares situation, it's important to confirm that you've explored other factoring options before concluding none are possible, especially if working over the reals.
Prime Polynomial
A prime polynomial is one that cannot be factored into the product of two non-constant polynomials over a given set of numbers; in this case, the real numbers. The polynomial \( a^2 + b^2 \) is an example, as it can't be factored with real coefficients. Prime polynomials are similar to prime numbers in arithmetic. Just as the number 7 can't be divided evenly by anything but 1 and itself, a prime polynomial remains "indivisible" in its current form over its number set. Determining if a polynomial is prime involves checking through all possible factor combinations and confirming that none work.
Polynomials
Understanding polynomials is essential for working with algebra. A polynomial is an expression made up of variables, coefficients, and involves operations of addition, subtraction, and multiplication. Polynomials come in various forms:
  • Constant polynomials like 3 or -5
  • Linear polynomials such as \( x + 2 \)
  • Quadratic polynomials like \( x^2 + 3x + 2 \)
When handling polynomials, factoring is a key operation used to simplify expressions or solve equations. In algebra, factoring a polynomial means expressing it as a product of its simpler building blocks, if possible. If no such simplification exists, like with a prime polynomial, the expression stays as it is.

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Most popular questions from this chapter

Factor using rational numbers. $$ x^{9}+y^{6} $$

Evaluate each expression. $$ -7^{2} $$

The following is a list of random factoring problems. Factor each expression. If an expression is not factorable, write "prime." See Examples 1-5. $$ 25 x^{2}-16 y^{2} $$

Factor. $$ 3 x^{4} y^{2}-29 x^{2} y+56 $$

The following is a list of random factoring problems. Factor each expression. If an expression is not factorable, write "prime." See Examples 1-5. $$ 8 x^{4}-8 $$
See all solutions

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