In Exercises 67–82, find each product. (x−y)(x 2 +xy+y 2 )

Hey everyone here we are asked to find the product of the given expression of the quantity of X times Y squared plus two X plus y squared times the quantity of one minus X. Now, all we have to do here is just multiply each term together. So we can start by multiplying these first three terms by one. So X times Y squared times one is just X times Y squared. Next we have two X times one which is just two X. And finally Y squared times one, which is just Y squared. Now we can just repeat this same process and multiply these three terms by now negative X. And so negative X times X times Y squared gives us negative X squared times Y squared negative X times two X. Gives us negative two X squared and finally negative X times Y squared is just negative X times Y squared. Now all we have to do is simplify what we have. So we can see that the x times y squares cancel. And this leaves us a final answer of negative X squared times Y squared minus two X squared plus Y squared plus two X. Which is our answer here for choice B Thanks for watching. I hope you found this video helpful and I'll see you again next time

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1:27 minutes

Problem 77

Textbook Question

In Exercises 67–82, find each product. (x−y)(x²+xy+y²)

Verified step by step guidance

Step 1: Recognize that the problem involves multiplying two expressions: \((x - y)\) and \((x^2 + xy + y^2)\). This is a straightforward application of the distributive property.

Step 2: Apply the distributive property by multiplying \((x - y)\) with each term in \((x^2 + xy + y^2)\). This means you will distribute \(x\) and \(-y\) across \((x^2 + xy + y^2)\).

Step 3: First, distribute \(x\) to each term in \((x^2 + xy + y^2)\): \(x \cdot x^2 + x \cdot xy + x \cdot y^2\), which simplifies to \(x^3 + x^2y + xy^2\).

Step 4: Next, distribute \(-y\) to each term in \((x^2 + xy + y^2)\): \(-y \cdot x^2 - y \cdot xy - y \cdot y^2\), which simplifies to \(-x^2y - xy^2 - y^3\).

Step 5: Combine all the terms from Step 3 and Step 4: \(x^3 + x^2y + xy^2 - x^2y - xy^2 - y^3\). Then, simplify by combining like terms to get the final expression.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Factoring and Distributing

Factoring involves breaking down expressions into simpler components, while distributing refers to applying the distributive property to multiply terms. In this case, we need to distribute (x - y) across the polynomial (x^2 + xy + y^2) to find the product. Understanding how to apply these operations is crucial for simplifying algebraic expressions.

Polynomial Multiplication

Polynomial multiplication involves multiplying two or more polynomials together, which requires combining like terms and applying the distributive property. Each term in the first polynomial must be multiplied by each term in the second polynomial. This concept is essential for solving the given expression, as it allows us to systematically find the resulting polynomial.

Combining Like Terms

Combining like terms is the process of simplifying an expression by adding or subtracting terms that have the same variable raised to the same power. After distributing and multiplying the polynomials, it is important to identify and combine these like terms to arrive at the final simplified expression. This step is key to ensuring the answer is presented in its simplest form.

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