Problem 84 - Introductory Algebra for College Students

Question

Multiply: \(\quad(x-y)\left(x^{2}+x y+y^{2}\right)\) (Section \(5.2,\) Example 7 )

Step-by-Step Solution

Answer

The result of the multiplication is: \(x^3 + x^2y + xy^2 -yx^2 - y^2x - y^3\)

Step 1: Distribute \(x\) from the binomial

First, multiply the \(x\) from the binomial \((x-y)\) with every term in the trinomial \((x^2 + xy + y^2)\). This gives \(x \cdot x^2 = x^3\), \(x \cdot xy = x^2y\), and \(x \cdot y^2 = xy^2\).

Step 2: Distribute \(-y\) from the binomial

Next, multiply the \(-y\) from \((x-y)\) with every term in the trinomial. This gives \(-y \cdot x^2 = -yx^2\), \(-y \cdot xy = -y^2x\), and \(-y \cdot y^2 = -y^3\).

Step 3: Combine Like Terms and Write the Product

Now, add together all the individual products obtained from Steps 1 and 2 to get the final result: \(x^3 + x^2y + xy^2 -yx^2 - y^2x - y^3\). Here, there are no like terms to combine, so this is the final answer.

Key Concepts

Distributive Property

The distributive property is a fundamental algebraic principle that helps in simplifying expressions and equations.
This property states that a term outside a set of parentheses can be multiplied by each term inside the parentheses.
This multiplication distributes the outer term across all terms within the bracket, succinctly written as \((a + b)(c + d) = a(c + d) + b(c + d)\).

• In our exercise, the binomial \(x - y\) is distributed over the trinomial \(x^2 + xy + y^2\).
• The first step involves multiplying each term of the trinomial by the first term of the binomial \(x\), then repeating the process with the second term \(-y\).
• This yields the terms: \(x^3, x^2y, xy^2\) and \(-yx^2, -y^2x, -y^3\).

Understanding the distributive property equips students with a versatile tool for simplification, making seemingly complicated expressions more manageable.

Binomial and Trinomial Multiplication

Multiplying a binomial by a trinomial involves using the distributive property to cover every term present in both expressions.
It's about pairing each term from the binomial with all individual terms from the trinomial, which can seem tricky at first.
In this example:

• The binomial \(x - y\) has two terms: \x\ and \(-y\).
• The trinomial \(x^2 + xy + y^2\) consists of: \(x^2, xy, and y^2\).
• Each term of the binomial multiplies each term of the trinomial creating new expressions as shown: \(x^3, x^2y, xy^2, -yx^2, -y^2x, -y^3\).

The key is to ensure that each term from the binomial is distributed and multiplied correctly with each term in the trinomial. Once all multiplications are carried out, the next step involves simplifying the result.

Combining Like Terms

Once each term has been multiplied and distributed, it's time to combine like terms, if any.
Like terms are those which have the same variable and corresponding power.
This aids in further simplifying the expression to its cleanest form.

• In our given problem, once the distribution yields: \(x^3 + x^2y + xy^2 - yx^2 - y^2x - y^3\), we search for terms that are alike.
• Typically, combining involves adding or subtracting coefficients of like terms.
• However, in this equation, all resulting terms: \(x^3, x^2y, xy^2, -yx^2, -y^2x, -y^3\) are unique, leaving our expression as simplified as possible without further action needed.

By effectively combining like terms, you ensure that the algebraic expression is as simplified and manageable as it can be, aligning with the goal of clarity in algebraic manipulation.