Problem 82 - Elementary Algebra

Question

For the following problems, simplify each of the algebraic expressions. $$ 3 x y^{2}(4 x y+5 y)+2 x y^{3}+6 x^{2} y^{3}+4 y^{3}-12 x y^{3} $$

Step-by-Step Solution

Answer

Question: Simplify the following algebraic expression: $$3xy^2(4xy+5y)+2xy^3+6x^2y^3+4y^3-12xy^3$$. Answer: The simplified algebraic expression is $$18x^2y^3 + 5xy^3 + 4y^3$$.

Step 1: Identify like terms

In the given expression, $$3xy^2(4xy+5y)+2xy^3+6x^2y^3+4y^3-12xy^3$$, we must first identify the like terms.

Step 2: Distribute the first term

We have one term that contains another expression inside parentheses. We need to distribute the term to the terms inside parentheses: $$3xy^2(4xy+5y)$$. Distributing gives us $$12x^2y^3 + 15xy^3$$.

Step 3: Rewrite the expression with the new terms

Now rewrite the expression by replacing the distributed term: $$12x^2y^3 + 15xy^3 + 2xy^3 + 6x^2y^3 + 4y^3 - 12xy^3$$.

Step 4: Combine like terms

Combine any like terms by adding or subtracting their coefficients: - Combine $$12x^2y^3$$ and $$6x^2y^3$$, which gives $$18x^2y^3$$. - Combine $$15xy^3$$, $$2xy^3$$, and $$-12xy^3$$, which gives $$5xy^3$$.

Step 5: Write the simplified expression

Finally, write down the simplified expression: $$18x^2y^3 + 5xy^3 + 4y^3$$.

Key Concepts

Polynomial Distribution

Understanding the concept of polynomial distribution is critical when dealing with algebraic simplification. Polynomial distribution involves applying the distributive property to multiply a monomial or a polynomial by another polynomial. In simpler terms, we need to multiply every term in one polynomial by every term in another. This is exactly what we apply when we encounter expressions like 3xy^2(4xy + 5y).

Essentially, you are distributing the 3xy^2 across the 4xy + 5y, resulting in 12x^2y^3 + 15xy^3. This process requires close attention to the exponents on the variables to make sure we combine them correctly when multiplying. For example, when we multiply xy^2 by xy, we add the exponents of like bases according to the laws of exponents, giving us x^2y^3 as a result.

Once the distribution is complete, it paves the way for us to combine like terms and further simplify the expression, leading to a more refined and concise result.

Combining Like Terms

Combining like terms is a fundamental approach in algebra that helps simplify expressions and solve equations. The process involves grouping terms that have the same variables raised to the same powers, and then combining them by adding or subtracting their coefficients. In our example 12x^2y^3 + 15xy^3 + 2xy^3 + 6x^2y^3 + 4y^3 - 12xy^3, we identify and combine terms with the same variable factors.

Specifically:

• 12x^2y^3 and 6x^2y^3 are combined because they both have x^2y^3 (resulting in 18x^2y^3).
• Similarly, 15xy^3, 2xy^3, and -12xy^3 form a group since they all have xy^3 (totally to 5xy^3).

The term 4y^3 has no like terms, so it remains unchanged in the expression.

After combining like terms, the expression is significantly simplified, which makes it easier to work with for further operations such as evaluating, graphing, or incorporating into other algebraic expressions.

Algebraic Simplification

Algebraic simplification is the process of making an algebraic expression as straightforward as possible. This simplification includes several steps and techniques such as removing parentheses, combining like terms, and reducing fractions to their simplest form, if present. The main goal is to make the expression easier to understand and work with. Using our initial complex expression, after applying polynomial distribution and combining like terms, we achieve the simplified form: 18x^2y^3 + 5xy^3 + 4y^3.

Such simplification not only makes the equation neater but it also makes it easier to evaluate for given variable values, compare to other expressions, or even integrate into a larger problem. Hence, mastering algebraic simplification is vital for anyone looking to excel in algebra and beyond, as these skills are routinely applied across various fields of math and science. With practice, identifying opportunities for simplification becomes more intuitive, allowing for quick and accurate problem solving.