Problem 101 - Algebra A Combined Function

Question

What binomial multiplied by \((x-6)\) gives the difference of two squares?

Step-by-Step Solution

Answer

The binomial is \(x + 6\).

Step 1: Understanding Binomials and Difference of Squares

Recall that a difference of two squares is of the form \(a^2 - b^2\) and can be factored as \((a - b)(a + b)\). We have one of the binomials as \((x - 6)\), so we need to identify the other binomial.

Step 2: Express Difference of Squares Including \(x-6\)

Assume the form \((x - a)(x + a)\) for the difference of two squares, where \(x - a = x - 6\). This suggests that \(a = 6\). Thus, the multiplication is \((x - 6)(x + 6) = x^2 - 36\).

Step 3: Identifying the Required Binomial

Since multiplying \((x - 6)\) by \((x + 6)\) yields \(x^2 - 36\), the binomial we need is \((x + 6)\).

Key Concepts

Binomial Multiplication

When we refer to binomial multiplication, we mean multiplying two algebraic expressions, each with two terms. Think of a binomial as an expression like \( (a + b) \) or \( (c - d) \). Binomial multiplication is the process of distributing each term in the first binomial to every term in the second. This process is more commonly remembered through the acronym FOIL: First, Outer, Inner, Last. However, in the context of the given problem, we are multiplying in a manner that specifically creates a difference of squares.
For example, consider the multiplication of \((x - 6)(x + 6)\). Using FOIL:

• First: Multiply the first terms, \(x \times x = x^2\).
• Outer: Multiply the outer terms, \(x \times 6 = 6x\).
• Inner: Multiply the inner terms, \(-6 \times x = -6x\).
• Last: Multiply the last terms, \(-6 \times 6 = -36\).

When you combine like terms, the \(6x\) and \(-6x\) cancel each other, leaving us with the simplified form \(x^2 - 36\). This is the difference of squares formula at work, illustrating the beauty and symmetry in algebraic multiplication.

Factoring

Factoring is the reverse process of multiplication. It's about breaking down complex algebraic expressions into simpler ones that, when multiplied together, will give back the original expression. In the exercise, we deal with the difference of squares, a type of expression that is ideal for factoring.
The difference of squares follows a specific pattern. It's of the form \(a^2 - b^2\). Such an expression can always be factored into \((a - b)(a + b)\). For example, for \(x^2 - 36\), recognize that 36 is a perfect square, \(6^2\), so \(x^2 - 36\) becomes \((x - 6)(x + 6)\) when factored. This form is significant because it exemplifies how factors unfold in pairs to give original expressions.
Recognizing these patterns is crucial in algebra because it allows us to simplify equations and solve a variety of problems more easily.

Algebraic Expressions

An algebraic expression is a mathematical phrase that can contain numbers, variables (like \(x\) or \(y\)), and operators (such as plus and minus). They're the building blocks for defining relationships and solving equations in algebra.
In the context of the given problem, the expressions \(x - 6\) and \(x + 6\) are binomial algebraic expressions. They represent values that vary as the variable \(x\) changes. The manipulation of these expressions through operations such as multiplication and factoring allows us to reshape and solve for the unknowns.
The process of solving algebraic expressions often involves identifying patterns and applying appropriate formulas, like the difference of squares, to break them down into more easily manageable parts. Understanding and being able to manipulate these expressions is an essential skill in algebra, as it lays the foundation for understanding more complex mathematical concepts.