Question

Multiple Choice Choose the best description of \(x^{2}-64\) (a) Prime (b) Difference of two squares (c) Difference of two cubes (d) Perfect Square

Short answer

The best description is (b) Difference of two squares.

Step-by-step solution

- Understand the Expression

Analyze the given expression, which is \[x^{2} - 64\]. Note that it is a subtraction involving a square term.

- Identify if It’s a Prime Number

Check if \[x^{2} - 64\] cannot be factored into simpler terms. A prime number cannot be factored, but polynomial expressions can be factored.

- Identify the Difference of Two Squares

Recognize that \[x^{2}\text{ and }64 \text{ are both perfect squares}\]. Rewrite \[64\] as \[8^{2}\]. Thus, \[x^{2} - 64 = x^{2} - 8^{2}\]. This represents the difference of two squares, which can be factored as: \[(x + 8)(x - 8)\].

- Verify It is Not a Difference of Two Cubes

Determine if the expression can be written as \[a^{3} - b^{3}\] format. Since neither \[x^{2} \text{ nor }64\] are cubes, this option is incorrect.

- Identify If It’s a Perfect Square

Check if \[x^{2} - 64\] can be written as \[(a + b)^{2}\]. As 64 is not a square of another binomial, this option is also incorrect.

Key concepts

Difference of Squares

The 'difference of squares' is a powerful algebraic concept. It takes the form a^{2} - b^{2} = (a + b)(a - b).In our exercise, the expression is x^{2} - 64.Recognize that both x^{2} and 64 are squares. Rewrite 64 as 8^{2}. So, our expression becomes x^{2} - 8^{2}.We apply the difference of squares rule: (x + 8)(x - 8).This means x^{2} - 64decomposes into two simpler linear terms.

Factoring Polynomials

Factoring polynomials is a fundamental skill in algebra. It involves breaking down a complex polynomial into simpler factors. In the given example, we start with the polynomial x^{2} - 64.This is a second-degree polynomial. Factoring it involves recognizing patterns like the difference of squares. Once recognized, rewrite this as (x + 8)(x - 8).Factoring simplifies solving equations, finding roots, and understanding polynomial properties.

Identifying Polynomial Forms

It's crucial to identify the form of a polynomial for solving and factoring. Polynomials come in various forms like:

• Difference of Two Squares: a^{2} - b^{2} = (a + b)(a - b)
• Perfect Squares: (a + b)^{2} or (a - b)^{2}
• Difference of Two Cubes: a^{3} - b^{3} = (a - b)(a^{2} + ab + b^{2})

In our case, we identified the difference of squares as x^{2} - 8^{2} = (x + 8)(x - 8).Recognizing forms is step one in tackling polynomial problems.

Prime Polynomial

A prime polynomial cannot be factored further over the set of real numbers. For example, x^{2} + 1is prime because it does not simplify into linear factors. For x^{2} - 64,we checked if it's prime by testing factoring options. Since x^{2} - 8^{2}equals (x + 8)(x - 8),it's not prime. Prime polynomials are rare and crucial for higher algebra understanding.