Problem 78 - College Algebra Essentials

Question

In Exercises 67–82, find each product. $$(x+y)\left(x^{2}-x y+y^{2}\right)$$

Step-by-Step Solution

Answer

The product of \(x+y\) and \(x^{2}-xy+y^{2}\) is \(x^{3}+x^{2}y-xy^{2}+y^{3}\).

Step 1: Distributing terms

Distribute \(x\) across \(x^{2}-x y+y^{2}\) to get \(x^{3}-x^{2}y+xy^{2}\). Similarly, distribute \(y\) across \(x^{2}-x y+y^{2}\) to get \(yx^{2}-y^{2}x+y^{3}\).

Step 2: Combining like terms

Combine the similar terms from step 1 to simplify the polynomial. \(x^{3}-x^{2}y+xy^{2}\) and \(yx^{2}-y^{2}x+y^{3}\) upon combining similar terms give \(x^{3}-x^{2}y+2x^{2}y- y^{3}+ y^{3}\).

Step 3: Final Simplification

The polynomial simplifies to \(x^{3}+x^{2}y-xy^{2}+y^{3}\) after combining the similar terms.

Key Concepts

Distributive Property

When you hear about the distributive property in mathematics, think about sharing. Just like when you distribute candy among friends, in math, you distribute terms in an equation across other terms to make the problem easier to solve. In polynomial multiplication, this property allows you to multiply a single term by each term inside a set of parentheses.

Imagine you want to multiply \((x+y)\) with \((x^2 - xy + y^2)\). Use the distributive property by distributing the first term,\(x\), to each term inside the parenthesis \((x^2 - xy + y^2)\). You will get:

• \(x \cdot x^2 = x^3\)
• \(x \cdot (-xy) = -x^2y\)
• \(x \cdot y^2 = xy^2\)

Next, you apply the distributive property to the second term, \(y\), the same way:

• \(y \cdot x^2 = yx^2\)
• \(y \cdot (-xy) = -y^2x\)
• \(y \cdot y^2 = y^3\)

By distributing all terms, you set up the equation to combine similar parts next.

Combining Like Terms

With the result of applying the distributive property on the problem, you now have a list of terms. The next step in simplifying polynomials involves combining like terms. Like terms are terms that have the exact same variables raised to the same powers. For example, \(x^2y\) and \(yx^2\) are like terms because they involve the same variables raised to the same powers.

After distribution, your polynomial looks like this:\[x^3 - x^2y + xy^2 + yx^2 - y^2x + y^3\] Now, look closely and group all of the like terms:

• \(x^3\): appears only once
• \(-x^2y\) and \(yx^2\): both are \(-x^2y\) and \(+x^2y\), combine to give \(0\)
• \(xy^2\) and \(-y^2x\): combine them for simplification
• \(y^3\): appears only once

Combining these terms simplifies your polynomial even further by focusing only on the surviving terms.

Simplifying Polynomials

Once you've combined all like terms, the last step in polynomial multiplication is to simplify the polynomial by writing it in standard form, with terms arranged in a specific way based on their degree.

In our example, after combining like terms, we find the expression simplified to:\(x^3 + x^2y - xy^2 + y^3\)
Here's what you need to check when simplifying:

• All similar terms are combined
• The polynomial is arranged from the highest degree to the lowest

This polynomial is already in simplified form because the terms are ordered, and each has a unique combination of powers.Simplifying polynomials helps reach a clean, concise expression, making them easier to work with for further calculations or applications.