Problem 57 - Algebra and Trigonometry

Question

Factor the expression completely. $$ (a+b)^{2}-(a-b)^{2} $$

Step-by-Step Solution

Answer

The expression factors to \(4ab\).

Step 1: Recognize the Difference of Squares Formula

The expression \((a+b)^{2}-(a-b)^{2}\) resembles the difference of squares formula, which is \(x^2 - y^2 = (x-y)(x+y)\). The terms \((a+b)^2\) and \((a-b)^2\) are perfect squares.

Step 2: Identify X and Y

Let \(x = (a+b)\) and \(y = (a-b)\). So the expression \((a+b)^{2}-(a-b)^{2}\) becomes \(x^2 - y^2\).

Step 3: Apply the Difference of Squares Formula

Use the formula \(x^2 - y^2 = (x-y)(x+y)\) and substitute \(x = (a+b)\) and \(y = (a-b)\). This results in \[((a+b)-(a-b))((a+b)+(a-b))\]

Step 4: Simplify Each Factor

Simplify the expressions in each factor: 1. \((a+b)-(a-b)\) simplifies to \(a+b-a+b = 2b\).2. \((a+b)+(a-b)\) simplifies to \(a+b+a-b = 2a\).

Step 5: Write the Final Factored Expression

Combine the simplified factors from Step 4 to write the factored expression: \(2b \cdot 2a = 4ab\). Therefore, the complete factored form of the given expression is \(4ab\).

Key Concepts

Factoring Techniques

Factoring is an important technique in algebra that involves rewriting an expression as a product of its factors. It's like breaking down a number into its constituents so that it can be multiplied back to form the original number. Factoring can simplify complex expressions and solve equations efficiently.

• One common factoring technique is the **Difference of Squares**, which applies to expressions in the form of \(a^2 - b^2\). It factors down to \((a - b)(a + b)\).
• Recognizing patterns is key. In difference of squares, both terms must be perfect squares.
• Factoring requires practice to spot these patterns and apply the correct method quickly.

Breaking down complex algebraic expressions using factoring techniques helps in solving equations and understanding the relationships between terms.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations. They represent mathematical relationships and can include operations such as addition, subtraction, multiplication, and division. Let's explore their components and features.

• **Variables** are symbols used to represent numbers, commonly letters like \(a\), \(b\), \(x\), or \(y\).
• **Coefficients** are numbers that multiply the variables. In \(3x\), 3 is the coefficient.
• **Terms** are the separate components of an expression, like \(a\) and \(b\) in \(a+b\).
• **Operations** show how terms are combined, using symbols like \(+\), \(-\), \(\times\), or \(\div\).

Algebraic expressions are foundational in learning algebra, enabling problem-solving and understanding relationships between variables. They serve as the building blocks of equations and functions, allowing for manipulations and transformations like factoring or expanding.

Squared Binomials

A squared binomial is an expression of the form \((a + b)^2\) or \((a - b)^2\). Expanding these expressions involves using the distributive property.

• The formula for squaring a binomial \((a + b)^2\) is \(a^2 + 2ab + b^2\). For a negative sign, \((a - b)^2\) is \(a^2 - 2ab + b^2\).
• Squaring a binomial essentially means multiplying the term by itself.

Understanding squared binomials helps in recognizing patterns in more complex algebraic expressions. These patterns are particularly useful when applying factoring techniques, such as the difference of squares. By identifying perfect square terms, students can easily execute these operations and simplify expressions.