Problem 102 - Algebra A Combined Function

Question

What binomial multiplied by \((5+y)\) gives the difference of two squares?

Step-by-Step Solution

Answer

The binomial is \((5-y)\).

Step 1: Understand the Difference of Squares Pattern

The difference of squares is a special pattern in algebra given by the equation \(a^2 - b^2 = (a + b)(a - b)\). We need to have the product of two binomials match this equation.

Step 2: Identify Variables in the Equation

Compare the pattern \((a+b)(a-b)\) with our given binomial \((5+y)\); hence we can equate it to either \((a+b)\) or \((a-b)\) and find what the other factor should be.

Step 3: Match Binomials

For the expression \((5+y)(5-y)\), this follows the pattern \((a+b)(a-b)\) where \(a = 5\) and \(b = y\). Thus, the binomial that multiplies \((5+y)\) to give a difference of squares is \((5-y)\).

Step 4: Verify the Expression

Multiply \((5+y)(5-y)\) to check if it results in a difference of squares: \((5+y)(5-y) = 5^2 - y^2 = 25 - y^2\), which is indeed a difference of squares.

Key Concepts

Binomials

In algebra, a binomial is an expression that consists of two terms. These terms are typically joined by addition or subtraction. For example, in \((5+y)\), the two terms are 5 and y. Binomials are special because they allow us to observe patterns and simplify expressions easily.

• Binomials contain exactly two distinct terms.
• They form the basic building blocks for more complex algebraic expressions.
• Understanding binomials is crucial for simplifying expressions and solving equations.

Recognizing binomials in algebraic expressions can help in applying algebraic techniques more efficiently, such as factoring, expanding, or using special products. This is foundational for further exploration into polynomials and algebraic identities.

Algebraic Patterns

Algebraic patterns help us understand how certain expressions relate or behave when manipulated. One of the key algebraic patterns is the difference of squares, represented by the equation \(a^2-b^2 = (a+b)(a-b)\). This pattern reveals that the product of a sum and a difference of the same two terms results in the subtraction of their squares.

• The difference of squares pattern helps simplify complex algebraic expressions.
• Recognizing this pattern allows us to factor expressions quickly.
• It's a powerful tool useful across different areas of algebra and calculus.

By identifying algebraic patterns, we can make strategic decisions about how to solve algebraic problems, switch between forms, or verify solutions with greater ease and confidence.

Multiplying Binomials

Multiplying binomials means expanding the product of two binomial expressions. This process is often accomplished using the FOIL method, which stands for First, Outer, Inner, Last, referring to the pairs of terms you multiply together from each binomial.
Consider multiplying two binomials, such as \((a+b)(c+d)\). Using the FOIL method, we multiply these pairs:

• **First Terms:** \(a \times c\)
• **Outer Terms:** \(a \times d\)
• **Inner Terms:** \(b \times c\)
• **Last Terms:** \(b \times d\)

Combined, these give us the expanded expression \(ac + ad + bc + bd\). When dealing with the difference of squares, the middle terms effectively cancel each other out, leaving just \(a^2 - b^2\). This demonstrates the utility of understanding how multiplying binomials intersect with algebraic patterns to simplify expressions.