Problem 76 - Elementary Algebra

Question

Simplify the algebraic expressions for the following problems. $$ 4 x^{2} y^{2}(2 x-3 y-5)-16 x^{3} y^{2}-3 x^{2} y^{3} $$

Step-by-Step Solution

Answer

Question: Simplify the algebraic expression: $$4x^2y^2(2x - 3y - 5) - 16x^3y^2 - 3x^2y^3$$ Answer: The simplified algebraic expression is $$-8x^3y^2 - 15x^2y^3 - 20x^2y^2$$.

Step 1: Distribute the terms inside the parentheses

Multiply the term outside the parentheses by each term inside the parentheses, then apply the exponent rules: $$ 4x^2y^2(2x) - 4x^2y^2(3y) - 4x^2y^2(5) - 16x^3y^2 - 3x^2y^3 \\ = 8x^3y^2 - 12x^2y^3 - 20x^2y^2 - 16x^3y^2 - 3x^2y^3 $$

Step 2: Combine like terms

Combine the terms having the same variable and exponent to simplify the expression: $$ (8x^3y^2 - 16x^3y^2) + (-12x^2y^3 - 3x^2y^3) - 20x^2y^2 \\ = -8x^3y^2 - 15x^2y^3 - 20x^2y^2 $$

Step 3: Factor out common factors

In this case, there are no common factors amongst all three terms, so we cannot factor any further. Thus, the simplified algebraic expression is: $$ -8x^3y^2 - 15x^2y^3 - 20x^2y^2 $$

Key Concepts

Simplifying Expressions

Simplifying an algebraic expression involves breaking down the expression into its simplest form. The goal is to make the expression as straightforward and compact as possible. This can mean minimizing the number of terms or combining terms that are similar.

When simplifying, always:

• Evaluate any arithmetic operations first.
• Utilize algebraic properties such as the distributive, associative, and commutative laws.
• Combine like terms, which helps reduce the expression's complexity.

In the exercise, we started with a complex expression and used distribution and combining like terms to make it simpler. Each step drew us closer to the final, reduced algebraic statement.

Distribution in Algebra

Distribution is a handy tool in algebra that allows you to remove parentheses by multiplying each term within the parentheses by a term outside of it. This process relies on the distributive property, which states \( a(b + c) = ab + ac \).

In our exercise, we had the term \(4x^2y^2\) outside the parentheses, and we distributed it over \((2x - 3y - 5)\). This means multiplying \(4x^2y^2\) by each term inside the parentheses, giving us:

• \(4x^2y^2 \cdot 2x = 8x^3y^2\)
• \(4x^2y^2 \cdot (-3y) = -12x^2y^3\)
• \(4x^2y^2 \cdot (-5) = -20x^2y^2\)

Distribution helps transform complex expressions into simpler, equivalent expressions that can be more easily managed.

Combining Like Terms

Combining like terms is an essential technique in simplifying expressions. It involves adding or subtracting coefficients of terms that have the same variable and the same exponents.

In the simplified expression from our problem, like terms are those that share the same power of \(x\) and \(y\). For instance:

• \(8x^3y^2\) can be combined with \(-16x^3y^2\) because they share the same variables with identical powers.
• Similarly, \(-12x^2y^3\) can be combined with \(-3x^2y^3\).

Combining these terms effectively reduces the expression:

• \(8x^3y^2 - 16x^3y^2 = -8x^3y^2\)
• \(-12x^2y^3 - 3x^2y^3 = -15x^2y^3\)

This process of combining like terms aids in getting a cleaner and more manageable final expression.

Factoring in Algebra

Factoring is the process of breaking down an expression into a product of simpler factors. It can make complex expressions easier to work with, but not all expressions can be further factored.

In our exercise, after simplifying and combining like terms, we look to factor the expression. We typically look for common factors among terms. However, in the final expression:

• \(-8x^3y^2\)
• \(-15x^2y^3\)
• \(-20x^2y^2\)

There are no common factors among all three terms that allow for further factoring.

If common factors had been found, they would have been factored out to present the expression in its simplest multiplication form. In this case, the expression is already as simplified as can be without any common factors to extract.