Finding the Product of Two Polynomials
In high school mathematics, one of the fundamental operations we learn is the multiplication of polynomials. Knowing how to find the product of two polynomials is a crucial skill that will help you solve a wide range of algebraic problems.
In this article, we'll walk through the step-by-step process of finding the product of the two polynomials: 2x + 3y and x^2 - xy + y^2.
Step 1: Understand the Polynomials
The first step is to understand the structure of the two polynomials we're working with:
- The first polynomial is 2x + 3y, which has two terms: 2x and 3y.
- The second polynomial is x^2 - xy + y^2, which has three terms: x^2, -xy, and y^2.
Step 2: Multiply Each Term of the First Polynomial with Each Term of the Second Polynomial
To find the product of the two polynomials, we need to multiply each term of the first polynomial with each term of the second polynomial. This will give us a total of 6 products.
- 2x × x^2 = 2x^3
- 2x × (-xy) = -2x^2y
- 2x × y^2 = 2xy^2
- 3y × x^2 = 3x^2y
- 3y × (-xy) = -3xy^2
- 3y × y^2 = 3y^3
Step 3: Combine the Products
Now that we have all the individual products, we can combine them to get the final result:
(2x + 3y)(x^2 - xy + y^2) = 2x^3 - 2x^2y + 2xy^2 + 3x^2y - 3xy^2 + 3y^3
Simplifying the expression, we get:
(2x + 3y)(x^2 - xy + y^2) = 2x^3 - 2x^2y - xy^2 + 3x^2y - 3xy^2 + 3y^3
Therefore, the product of the two polynomials, 2x + 3y and x^2 - xy + y^2, is 2x^3 - 2x^2y - xy^2 + 3x^2y - 3xy^2 + 3y^3.
I hope this detailed explanation helps you understand the process of finding the product of two polynomials. If you have any further questions or need additional practice, feel free to reach out to me or consult other resources on the topic.