Problem 77 - 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 is \( x^{3} - y^{3} \)

Step 1: Distribute the First Term

Multiply the first term of the first expression \(x\) by each term in the second expression \(\left(x^{2}+x y+y^{2}\right)\). This results in: \( x*x^{2} + x*x*y + x*y^{2} = x^{3} + x^{2}y + xy^{2} \)

Step 2: Distribute the Second Term

Then, multiply the second term of the first expression \(-y\) by each term in the second expression \(\left(x^{2}+x y+y^{2}\right)\). This results in: \( -y*x^{2} -y*x*y -y*y^{2} = -yx^{2} - yx^{2} - y^{3} \)

Step 3: Combine Like Terms

Now, combine the results obtained in Step 1 and Step 2: \( x^{3} + x^{2}y + xy^{2} -yx^{2} - yx^{2} - y^{3} = x^{3} - y^{3}\)

Step 4: Final Answer

Carefully combining all terms gives the final result as \( x^{3} - y^{3}\)

Key Concepts

Multiplying Polynomials Step by Step

Understanding how to multiply polynomials is a fundamental skill in algebra. The process involves a few clear steps that, when followed in order, make the task manageable. The key here is systematic multiplication.

Let's break it down using the example from the exercise \( (x-y)(x^{2}+xy+y^{2}) \). The first step is to take each term from the first polynomial and multiply it by each term of the second polynomial. This is essentially applying the distributive property twice - once for \( x \) and once for \( -y \). It is important to keep track of signs; a positive times a positive is a positive, a positive times a negative is a negative, and a negative times a negative is a positive.

Once you have multiplied each term, you will end up with a longer polynomial that may contain like terms. These are terms that have the same variables raised to the same powers. The final step is simplifying by combining like terms, which often results in a polynomial of lower degree than might at first be expected.

Distributive Property

The distributive property is the cornerstone of polynomial multiplication. It states that \( a(b+c) = ab + ac \), which is applied when a single term is being multiplied by a binomial or a larger polynomial.

In our example, when we distribute \( x \) over \( x^{2}+xy+y^{2} \) and \( -y \) over the same trinomial, we apply this property. Every term in the brackets must be multiplied by the term outside the bracket, one at a time, ensuring not to miss any combinations. It's critical to align this step carefully to avoid errors in sign or coefficient. This is also where students can utilize a methodical approach, such as FOIL (First, Outer, Inner, Last) when multiplying binomials or a similar systematic practice for polynomials with more terms.

Combining Like Terms

After multiplication, combining like terms is what helps us to simplify polynomials to their most reduced form.

Like terms are terms within an algebraic expression that have the same variables and powers. For instance, \( x^{2}y \) and \( -yx^{2} \) are like terms because they contain the same variables raised to the same powers, and therefore can be combined. When combining, add or subtract the coefficients (the numbers in front of the variables) and leave the variable part unchanged.

The exercise provided simplified to \( x^{3} - y^{3} \) because all the intermediate terms involving \(xy \) cancelled each other out. Recognizing opportunities to combine like terms is essential for simplifying polynomials correctly. Developing a careful and thorough approach ensures accuracy in solving and simplifying polynomial expressions.