Problem 52 - College Algebra

Question

For the following exercises, find the sum or difference. $$(x+y)\left(x^{2}-x y+y^{2}\right)$$

Step-by-Step Solution

Answer

The sum is \(x^3 + y^3\).

Step 1: Expand the Expression

We need to multiply each term in the binomial \(x+y\) by each term in the trinomial \(x^2 - xy + y^2\). This is done using the distributive property.

Step 2: Multiply the First Term

First, distribute \(x\) across the trinomial: \(x(x^2 - xy + y^2) = x^3 - x^2y + xy^2\).

Step 3: Multiply the Second Term

Next, distribute \(y\) across the trinomial: \(y(x^2 - xy + y^2) = yx^2 - y^2x + y^3\).

Step 4: Combine the Results

Add the two results from Step 2 and Step 3 together: \(x^3 - x^2y + xy^2 + yx^2 - y^2x + y^3\).

Step 5: Simplify the Expression

Combine like terms: - \(x^3\) is alone.- Combine \(-x^2y\) and \(yx^2\) to get \(-x^2y + x^2y = 0\).- Combine \(xy^2\) and \(-y^2x\) to get \(xy^2 - xy^2 = 0\).- \(y^3\) is alone. The simplified expression is \(x^3 + y^3\).

Key Concepts

Distributive Property

The distributive property is an essential concept in algebra, and it's a rule that allows you to multiply a single term by each term within a parenthesis. When you apply this property, you're essentially distributing or sharing one quantity among others. In the given exercise, the distributive property is used to multiply the binomial \((x+y)\) by each term in the trinomial \(x^2 - xy + y^2\). The steps of distribution are systematic:

• Take each term of the binomial \((x+y)\) and multiply it by the entire trinomial \((x^2 - xy + y^2)\).
• This involves calculating separately for \(x\) and \(y\), multiplying them by each term of the trinomial.

By following this process, the expression is expanded into individual terms, which lays the groundwork for further simplification.

Binomials

A binomial is a type of polynomial that includes exactly two distinct terms. In the exercise provided, \(x+y\) is the binomial. Binomials are quite straightforward, making them easy to handle in algebraic operations. They are often used in polynomial expansion, like in this problem, where they interact with trinomials. Understanding binomials is crucial because:

• They form the basis for building more complex polynomials.
• Multiplying binomials is a stepping stone to more advanced polynomial operations.
• They frequently appear in formulas and algebraic expressions you’ll encounter in algebra.

Working with binomials trains you to think about how algebraic expressions can be built, manipulated, and expanded.

Trinomials

Trinomials are another key part of polynomials, known for having exactly three terms. In our exercise, \(x^2 - xy + y^2\) is the trinomial. Trinomials appear often in algebra, especially in problems involving polynomial expansion and simplification. Here's what makes trinomials important:

• They help in understanding the complexity and structure of other polynomials.
• Trinomials can often be factored or expanded, which is crucial for solving algebraic equations.
• They test your ability to manage multiple algebraic operations systematically.

In the expansion steps of our problem, the trinomial shows how multiple terms interact when multiplied by a binomial, leading to opportunities to practice finding like terms.

Like Terms

Like terms are essential in simplifying algebraic expressions. They are terms that have the same variables raised to the same powers. In the solution, identifying like terms is crucial to achieve the simplified form of the polynomial expansion. When you combine like terms, you are effectively streamlining the expression by adding or subtracting the coefficients of these terms. The steps show:

• Combining terms such as \(-x^2y\) and \(yx^2\) because they are like terms despite their order.
• This simplification helps in reducing the expression to a more manageable form, such as \(x^3 + y^3\).
• Recognizing and combining like terms simplifies calculations, making further algebraic manipulations easier.

Being able to spot like terms is a practical skill you will use often in algebra, helping you to make expressions as simple as possible.