Problem 52 - Algebra A Combined Function

Question

Multiply. $$ (y-5)(y+5) $$

Step-by-Step Solution

Answer

(y-5)(y+5) = y^2 - 25

Step 1: Understand the Expression

To solve the multiplication \((y-5)(y+5)\), we recognize it as a product of two binomials of the form \((a-b)(a+b)\), which is a difference of squares.

Step 2: Identify the Formula

The difference of squares formula states: \((a - b)(a + b) = a^2 - b^2\). For our problem, identify \(a = y\) and \(b = 5\).

Step 3: Apply the Formula

Substitute the identified values into the formula: \(y^2 - 5^2\).

Key Concepts

Binomials

A binomial is a type of algebraic expression with exactly two terms. These terms are typically separated by either a plus or minus sign.
It is fundamental in algebra because many problems, like the one given, involve working with binomials.Examples of binomials include:

• \(x + 3\)
• \(a - b\)
• \((y - 5)(y + 5)\), which is a special binomial case known as the difference of squares.

When multiplying binomials, recognizing their structure allows us to apply specific formulas, such as the difference of squares, simplifying our calculations tremendously. Understanding binomials and their behavior through operations like multiplication is essential for mastering algebra.

Multiplying Polynomials

Multiplying polynomials includes working with expressions that have more than one term. In our exercise, we focused on multiplying two binomials that form a special case called a difference of squares.
Using formulas can simplify these operations.Multiple polynomial multiplication usually involves distributing each term in the first polynomial by each term in the second polynomial, but special cases like the difference of squares can be simplified:- The difference of squares formula states: \((a-b)(a+b) = a^2 - b^2\)- For our problem: identify \(a = y\) and \(b = 5\)By plugging these values into the formula, we efficiently solve the multiplication:

• First, compute \(a^2\) which is \(y^2\).
• Then, compute \(b^2\) which is \(5^2 = 25\).
• Subtract the second result from the first to obtain \(y^2 - 25\).

Recognizing and applying these patterns helps simplify what might otherwise be complex calculations.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operation symbols. They form the backbone of algebra and provide a way to represent mathematical relationships.
The expression from the exercise, \((y-5)(y+5)\), involves variables \(y\) and constants \(-5\) and \(+5\), representing full algebraic expressions.Key points about algebraic expressions include:

• Expressions can be simplified using algebraic rules and formulas like the difference of squares.
• They are flexible and allow for substitution to solve equations or evaluate expressions for specific variable values.
• Recognition of special patterns within expressions can provide shortcuts, such as quickly identifying the difference of squares pattern.

Through a solid grasp of algebraic expressions, one can move forward in solving more complex algebraic equations and engaging in advanced mathematics.