Problem 19 - Elementary Algebra

Question

Find the following products and simplify. $$ \left(2 x^{2} y^{3}+x y^{2}\right)\left(5 x^{3} y^{2}+x^{2} y\right) $$

Step-by-Step Solution

Answer

Question: Multiply and simplify the given polynomials: \((2x^2y^3 + xy^2)(5x^3y^2 + x^2y)\) Answer: The simplified product of the given polynomials is \(10x^5y^5 + 7x^4y^4 + x^3y^3\).

Step 1: Distribute the first term of the first polynomial

Distribute the first term of the first polynomial, \(2x^2y^3\), to both terms of the second polynomial: $$(2x^2y^3)(5x^3y^2) + (2x^2y^3)(x^2y)$$

Step 2: Perform the multiplication

Multiply the corresponding terms in both expressions: $$10x^5y^5 + 2x^4y^4$$

Step 3: Distribute the second term of the first polynomial

Now distribute the second term of the first polynomial, \(xy^2\), to both terms of the second polynomial: $$(xy^2)(5x^3y^2) + (xy^2)(x^2y)$$

Step 4: Perform the multiplication

Multiply the corresponding terms in both expressions: $$5x^4y^4 + x^3y^3$$

Step 5: Combine products from Steps 2 and 4

Now we will combine the products obtained in steps 2 and 4: $$10x^5y^5 + 2x^4y^4 + 5x^4y^4 + x^3y^3$$

Step 6: Combine like terms

Combine the like terms in the expression: $$10x^5y^5 + 7x^4y^4 + x^3y^3$$ The simplified product of the given polynomials is: $$10x^5y^5 + 7x^4y^4 + x^3y^3$$

Key Concepts

Distributive Property

Understanding the distributive property is crucial when multiplying polynomials. This property allows us to "distribute" one term over others in an expression. In simple terms, it means that each term outside the parenthesis is multiplied by every term inside the parenthesis.

For the given polynomials, \( (2x^2y^3 + xy^2)(5x^3y^2 + x^2y) \), we apply the distributive property twice:

• First, distribute \( 2x^2y^3 \) by multiplying it with each term inside \( 5x^3y^2 + x^2y \).
• Next, distribute \( xy^2 \) by doing the same multiplication for each term.

This step ensures that each term from the first polynomial is multiplied with each term from the second polynomial, leaving you with a complete set of products that can be simplified. Understanding how to effectively use the distributive property is key to achieving a correct solution.

Like Terms

When dealing with polynomial expressions, identifying and combining like terms is an essential skill. Like terms are terms that have the same variable components with identical exponents.

In the expression \( 10x^5y^5 + 2x^4y^4 + 5x^4y^4 + x^3y^3 \), notice:

• \(2x^4y^4\) and \(5x^4y^4\) are like terms because they share both the same variable bases and exponents.

Combine them to simplify the expression, resulting in \( 7x^4y^4 \). This step is crucial as it reduces the complexity of your final expression, approaching the solution in its simplest form.

Simplification

Simplification in polynomial multiplication involves transforming the expression into its most reduced form. This process is typically achieved through a mix of distribution, multiplication, and combining like terms.

Once you have applied the distributive property and identified all like terms, you are ready to simplify.Consider the expression \( 10x^5y^5 + 7x^4y^4 + x^3y^3 \):

• Check if there are any further like terms to combine, ensuring all terms in the expression have been addressed.
• Verify whether coefficients or variables can be further simplified or factored out.

The ultimate goal of simplification is to have a clear, concise expression that is easy to interpret and use in subsequent equations or calculations. Keeping this process orderly results in an accurate outcome in polynomial multiplication.