Problem 39 - Elementary and Intermediate Algebra

Question

Factor. If a polynomial can't be factored, write "prime." $$ x^{2}-16 $$

Step-by-Step Solution

Answer

The polynomial \(x^2 - 16\) factors into \((x - 4)(x + 4)\).

Step 1: Recognize the Form

The polynomial given is \(x^2 - 16\). Notice that this is a difference of squares, which takes the form \(a^2 - b^2\). In this case, \(a = x\) and \(b = 4\), since \(4^2 = 16\).

Step 2: Apply the Difference of Squares Formula

The difference of squares formula is \(a^2 - b^2 = (a - b)(a + b)\). Substitute \(a = x\) and \(b = 4\) into this formula to get:\[x^2 - 16 = (x - 4)(x + 4)\]

Step 3: Verify the Factorization

To confirm the factorization is correct, expand \((x - 4)(x + 4)\):\[(x - 4)(x + 4) = x^2 + 4x - 4x - 16 = x^2 - 16\]This confirms that the factorization is correct.

Key Concepts

Difference of Squares

The difference of squares is a specific algebraic expression where two square terms are subtracted from each other. It takes the form of \(a^2 - b^2\). For instance, the expression \(x^2 - 16\) is a difference of squares because it can be expressed as \((x)^2 - (4)^2\). This type of expression can be factored using a special formula:

• \(a^2 - b^2 = (a - b)(a + b)\)

This formula essentially says that the expression can be thought of as a product of two binomials: one formed by subtracting \(b\) from \(a\), and the other by adding \(b\) to \(a\). For example, using our given exercise, we have \(a = x\) and \(b = 4\), thereby writing:

• \(x^2 - 16 = (x - 4)(x + 4)\)

Understanding how to recognize and use the difference of squares formula can dramatically simplify your work on similar algebraic problems.

Factoring Techniques

Factoring is an essential process in algebra where expressions are simplified or rearranged into a product of factors. These factors can then be further utilized to solve equations or to make algebraic expressions more understandable. Several techniques are used for factoring, including:

• GCF (Greatest Common Factor): Factoring out the highest common factor shared by the terms in the expression.
• Grouping: Useful for polynomials with four terms, grouping involves rearranging terms into pairs that can be factored separately.
• Difference of Squares: As discussed, uses the form \(a^2 - b^2 = (a - b)(a + b)\).
• Trinomials: Factoring expressions typically in the format \(ax^2 + bx + c\).

The technique to use often depends on the specific form of the polynomial. By mastering various factoring strategies, you can effortlessly work through a range of algebraic problems.

Algebraic Expressions

An algebraic expression involves variables, numbers, and arithmetic operations like addition, subtraction, multiplication, and division. They serve as the foundation of algebra and are crucial for expressing real-world scenarios in a mathematical format. For example, \(x^2 - 16\) is an algebraic expression that we'll often factor for simplification or further analysis.Breaking down an algebraic expression into its components helps us better understand the problem and solve it:

• Terms: Individual parts of an expression, such as \(x^2\) or \(-16\).
• Coefficients: The numerical part of terms involving variables, e.g., the coefficient of \(x^2\) is 1.
• Constants: Numbers without variables, such as -16 in our example.

Learning to manipulate and simplify algebraic expressions using techniques like factoring is a key skill in mathematics, making it easier to handle complex equations and systems.