Problem 20 - Algebra for College Students

Question

Use the difference-of-squares pattern to factor each of the following. $$(x-1)^{2}-(x-8)^{2}$$

Step-by-Step Solution

Answer

The expression factors to \((2x-9)(7)\).

Step 1: Recognize the Difference of Squares Pattern

The expression \((x-1)^{2}-(x-8)^{2}\) is a difference of squares. The general form for a difference of squares is \(a^{2} - b^{2}\), which can be factored into \((a+b)(a-b)\).

Step 2: Identify \(a\) and \(b\) in the Expression

Identify \(a\) and \(b\) in the expression \((x-1)^{2}-(x-8)^{2}\). Here, \(a = (x-1)\) and \(b = (x-8)\).

Step 3: Apply the Factoring Formula

Using the difference of squares formula, substitute \(a\) and \(b\) into \((a+b)(a-b)\). This gives us:\[(x-1 + (x-8))(x-1 - (x-8))\]

Step 4: Simplify the Factored Expression

Simplify the terms inside the parentheses:\[= ((x-1)+(x-8))((x-1)-(x-8))\] Now,- First part: \((x-1) + (x-8) = x - 1 + x - 8 = 2x - 9\)- Second part: \((x-1) - (x-8) = x - 1 - x + 8 = 7\)So, the expression becomes \((2x - 9)(7)\).

Step 5: Write the Final Factored Form

Thus the factored form of \((x-1)^{2}-(x-8)^{2}\) is \((2x - 9)(7)\).

Key Concepts

Factoring Algebraic Expressions

Factoring algebraic expressions involves rewriting an expression as a product of its factors. This process is like determining the numbers or expressions that multiply to form the original expression. Factoring allows us to simplify algebraic problems and solve equations more easily.

When we factor algebraic expressions, we look for patterns or techniques to split the expression. They can be recognized by specific formulas such as the difference of squares, perfect square trinomials, or by simply pulling out common factors.

Using the difference-of-squares pattern requires recognizing an expression that is structured as the difference of two perfect squares, like \( a^2 - b^2 \). This expression can be refactored into the product \((a+b)(a-b)\).

If you understand factoring, you can streamline the solutions to algebraic expressions and can apply these methods in multiple contexts, from simplifying expressions to solving quadratic equations swiftly. Students should practice identifying common patterns to become comfortable with algebraic factoring.

Polynomial Factoring

Polynomials are algebraic expressions consisting of variables and coefficients combined using addition, subtraction, multiplication, and non-negative integer exponents. Factoring polynomials involves expressing the polynomial as a product of its factors, which are simpler, more manageable expressions.

Factoring polynomials is a crucial skill, especially for solving polynomial equations where you set the equation to zero and solve for the variable. When a polynomial is factored, it becomes easier to find its roots by setting each factor equal to zero.

There are several techniques for polynomial factoring:

• Finding the greatest common factor (GCF).
• Factoring by grouping.
• Recognizing common algebraic patterns like trinomials and difference of squares.
• Using formulas or identities.

Each technique applies to different types of polynomials, and with practice, you can learn to identify which method suits a given polynomial.

Ultimately, mastering polynomial factoring is foundational for higher-level algebra and calculus as it simplifies complex expressions, making them far more manageable to work with.

Algebraic Patterns

Algebraic patterns are formulas or techniques used to solve algebraic expressions efficiently. They help us recognize structures within problems and apply the right methods to simplify or solve them. The difference-of-squares is a frequently encountered algebraic pattern.

This pattern follows the format: \( a^2 - b^2 = (a+b)(a-b) \). Recognizing such patterns simplifies the process of factoring and can be particularly helpful for expressions that initially seem complicated.

Another example is the perfect square trinomial, which follows patterns like \( a^2 + 2ab + b^2 = (a+b)^2 \).

By learning these algebraic patterns, students can tackle more complex mathematical challenges. Recognizing these patterns saves time and reduces errors in calculations.

Through practice, students become adept at spotting these patterns and can apply them across various math problems, thus gaining efficiency and a deeper understanding of algebra.