Problem 92 - Algebra for College Students

Question

Determine the number of terms in the product of \((x+y)\) and \((a+b+c+d)\) without doing the multiplication. Explain how you arrived at your answer.

Step-by-Step Solution

Answer

The number of terms in the product is 8.

Step 1: Understand the problem

We are given two binomials, \((x+y)\) and \((a+b+c+d)\). The task is to find the number of terms in the expanded expression after multiplying them together, without actually performing the multiplication.

Step 2: Apply the distributive property

When multiplying two expressions, each term in the first expression must be multiplied by each term in the second expression. This is a direct application of the distributive property of multiplication over addition.

Step 3: Count the possible combinations

\((x+y)\) has 2 terms and \((a+b+c+d)\) has 4 terms. By the distributive property, each term from \((x+y)\) multiplies with each term from \((a+b+c+d)\). Therefore, the total number of combinations is given by the product of the number of terms in each binomial: \(2 \times 4 = 8\).

Step 4: Conclusion

As there are no repeated terms in the combination of terms, every combination results in a unique term. Thus, the number of unique terms in the final expanded expression is 8.

Key Concepts

Distributive Property

The distributive property is a fundamental principle in algebra used to simplify expressions and equations. It allows us to break down a multiplication operation across an addition operation. Think of it as spreading the multiplication over the entire sum. This is especially useful when dealing with expressions inside parentheses. For instance, when you have an expression like \((x+y)(a+b+c+d)\), each term in the first parenthesis must be multiplied by each term in the second parenthesis.

In more detail, the distributive property can be expressed as:

• \(a(b + c) = ab + ac\)
• This principle generalizes so that each term inside a pair of parentheses is multiplied by each term in the other pair.

By applying this property, we can fully expand expressions to easily identify all individual terms that make up a polynomial. In the exercise above, it is this property that allows us to determine there will be 8 terms in total without having to perform the entire multiplication, just by counting the combination of terms.

Binomials

A binomial is a type of polynomial that contains exactly two terms. For example, the expression \(x+y\) is a simple binomial. Binomials are a key concept in algebra because they form the basis for more complex expressions. Whenever you multiply two binomials, you’re creating a combination of terms through the distributive property.

Binomials follow the pattern \( (a + b) \), where both \(a\) and \(b\) can be any number, variable, or more complex expression. Multiplying binomials has its special cases, such as squaring a binomial, resulting in a recognizable distribution of terms including a middle term from cross-multiplication.

• Example: \( (a + b)^2 = a^2 + 2ab + b^2 \)
• This shows how each term combines at least once with every other term.

In the given exercise, \(x+y\) is a binomial, while the other expression \(a+b+c+d\) is technically not a binomial since it has four terms, but it functions similarly in the context of multiplication involving the distribution of terms.

Algebraic Expressions

An algebraic expression is a mathematical phrase involving numbers, variables, and operators such as addition, subtraction, multiplication, and division. In algebra, expressions are the building blocks used to model real-world situations and solve problems.

Every expression is composed of terms which can stand on their own or combine with others through operations. For example, in the exercise, \((x + y)\) and \((a + b + c + d)\) are both algebraic expressions as they consist of terms linked by the addition operation.

• Term: A single number, variable, or numbers and variables multiplied together.
• Expression: A collection of terms separated by addition or subtraction.

Algebraic expressions are endlessly applicable in mathematics, physics, engineering, and beyond, due to their versatility in representing various quantities. In our case, knowing how many terms an expression will have before actually solving it, like when using the distributive property, showcases the power of understanding these expressions deeply.