Problem 108 - Elementary Algebra

Question

For the following problems, perform the multiplications and combine any like terms. $$ (x+y)\left(2 x^{2}+3 x y+5 y^{2}\right) $$

Step-by-Step Solution

Answer

Question: Multiply the polynomials (x + y) and (2x^2 + 3xy + 5y^2) and simplify the result. Answer: 2x^3 + 5x^2y + 8xy^2 + 5y^3

Step 1: Distribute the terms

First, we need to distribute each term of the first polynomial \((x+y)\) to each term of the second polynomial \((2x^2 + 3xy + 5y^2)\).

Step 2: Multiply x with terms in the second polynomial

Start by multiplying \(x\) with each term in the second polynomial: $$ x(2x^2) + x(3xy) + x(5y^2) $$ This simplifies to: $$ 2x^3 + 3x^2y + 5xy^2 $$

Step 3: Multiply y with terms in the second polynomial

Now we multiply \(y\) by each term in the second polynomial: $$ y(2x^2) + y(3xy) + y(5y^2) $$ This simplifies to: $$ 2x^2y + 3xy^2 + 5y^3 $$

Step 4: Add the results of Steps 2 and 3

Now, we add the results of Steps 2 and 3 together: $$ (2x^3 + 3x^2y + 5xy^2) + (2x^2y + 3xy^2 + 5y^3) $$

Step 5: Combine like terms

Finally, we need to combine the like terms in the expression: $$ 2x^3 + (3x^2y + 2x^2y) + (5xy^2 + 3xy^2) + 5y^3 $$ This simplifies to: $$ 2x^3 + 5x^2y + 8xy^2 + 5y^3 $$ The final solution to the given problem is: $$ (x+y)\left(2 x^{2}+3 x y+5 y^{2}\right) = 2x^3 + 5x^2y + 8xy^2 + 5y^3 $$

Key Concepts

Distributive Property

The distributive property is a fundamental principle used in algebra to simplify expressions and solve equations. It allows you to multiply a single term by two or more terms inside a set of parentheses. This means you are distributing the multiplication over each term individually. In simpler terms, if you have an expression like \((a+b)(c+d)\), you'll apply the process

• Multiply \(a\) by both \(c\) and \(d\)
• Multiply \(b\) by both \(c\) and \(d\)

This operation results in new expressions that can later be combined. When applied correctly, the distributive property ensures all terms are properly multiplied, avoiding errors in calculations. Breaking down a complex polynomial expression into simpler components using this property makes handling larger problems much more manageable. This process helps create a foundation for further algebraic manipulations, such as combining like terms or simplifying equations.

Combining Like Terms

Combining like terms is a key step in simplifying an algebraic expression. Once you've used the distributive property to expand an expression, you might end up with several terms that can be combined. Like terms in algebra refer to terms that have the same variables raised to the same power.For example, in the expression \(3x^2y + 2x^2y\), both terms are like terms because they contain exactly the same variable parts: \(x^2y\). When combining like terms, you simply add or subtract their coefficients. Here's how it works:

• Match the terms: Identify terms with identical variable parts.
• Combine their coefficients: For the example above, add \(3\) and \(2\) to get \(5x^2y\).

Combining like terms helps to simplify an expression, making it easier to evaluate or solve. It's essential for reaching the most compact and informative version of an equation. This method ultimately helps in reducing complex expressions into a simpler form, which is a crucial skill in algebra.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations. They are used to represent values that can change, depending on what the variables represent. In the context of polynomial multiplication and simplification, algebraic expressions can include terms like \(2x^3\), \(5xy^2\), or \(8y^3\).Understanding the structure of algebraic expressions is essential:

• Variables: Represent unknown values and are typically noted as \(x\), \(y\), or other letters.
• Coefficients: Numbers that multiply variables, such as the \(3\) in \(3x\).
• Constants: Numbers without a variable part, like \(7\) or \(-3\).
• Operations: Include addition, subtraction, multiplication, or division which combine the terms.

These expressions form the basis of all algebraic equations and their manipulation is crucial for solving them. Whether expanding an expression using the distributive property or simplifying it by combining like terms, recognizing and understanding the parts of an algebraic expression is an essential step in mastering algebra.