Problem 45 - Elementary and Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

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

Step 1: 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.

Step 2: 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 \( a^2 - b^2 = (a - b)(a + b) \).

Step 3: 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.

Key Concepts

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.