Problem 53 - Algebra 2

Question

Find each product. Classify the result by number of terms. $$ (a-b)^{2}(a+b) $$

Step-by-Step Solution

Answer

The product is \( a^{3} - b^{3} \) which is a binomial.

Step 1: Apply the formula for the square of a binomial

Use the formula for the square of a binomial, \( (x - y)^{2} = x^{2} - 2xy + y^{2} \), to expand \( (a - b)^{2} \) which results in \( a^{2} - 2ab + b^{2} \) .

Step 2: Multiply the result by the third term

Now, multiply the expanded form \( a^{2} - 2ab + b^{2} \) by the third term \( (a + b) \) using the distributive property of multiplication over addition.

Step 3: Simplify the expression

Distribute each term in \( a^{2} - 2ab + b^{2} \) to each term in \( (a + b) \) and combine like terms to get the simplified result.

Step 4: Classify by the number of terms

Now that the expression is simplified, count the number of distinct terms to classify the result.

Key Concepts

Square of a Binomial

Understanding the square of a binomial is fundamental in polynomial multiplication. This term refers to an expression of the form \( (a+b)^2 \) or \( (a-b)^2 \) and essentially means multiplying the binomial by itself. Applying the formula \( (x+y)^2 = x^2 + 2xy + y^2 \) (or \( (x-y)^2 = x^2 - 2xy + y^2 \) if the sign is negative) helps one to quickly and accurately expand these expressions.

For instance, when asked to find the product \( (a-b)^2(a+b) \), we start with \( (a-b)^2 \) and apply the formula to get \( a^2 - 2ab + b^2 \), which is the expanded form of the square of the binomial \( (a-b) \) squared.

Distributive Property

The distributive property is a cornerstone concept that allows us to multiply a single term by each term inside a parenthesis. It states that \( a(b + c) = ab + ac \). When applying this property, each element of the first polynomial is multiplied by each element of the second.

By using this property on the expanded form \( a^2 - 2ab + b^2 \) and multiplying it by the binomial \( (a+b) \) in our exercise, we distribute each term of the trinomial across each term in the binomial, yielding terms such as \( a^3 \) and \( ab^2 \) before the expression is further simplified.

Simplify Expressions

Simplifying expressions involves combining like terms and reducing expressions to their simplest form. Like terms have the same variables raised to the same powers. After applying the distributive property, our next step is to group like terms to simplify the expression.

For the given problem, this simplification process reduces the number of terms by summing coefficients of like terms. The art of simplification makes the expression neater and often more useful, especially when it comes to subsequent calculations or solving equations.

Classify Polynomial Terms

Classification of polynomial terms is the final step in our exercise. Polynomials can be named based on the number of terms they contain: monomials (one term), binomials (two terms), and trinomials (three terms).

Once the expression is simplified, we count the number of distinct terms. The initial product of \( (a-b)^2(a+b) \) may result in several terms, but after combining like terms during simplification, the number of terms might decrease, leading to a new classification. For example, a four-term expression that simplifies to three terms would be classified as a trinomial.