Problem 67 - Elementary Algebra

Question

For the following problems, perform the multiplications and combine any like terms. $$ 2 x^{2} y\left(3 x^{2} y^{2}-6 x\right) $$

Step-by-Step Solution

Answer

Answer: The simplified expression is $$6x^4y^3 - 12x^3y$$.

Step 1: Distribute the term outside the parentheses

To perform the multiplication, we will distribute the term \(2x^2y\) to each term inside the parentheses. When multiplying two terms with the same base, add their exponents. $$ 2x^2y(3x^2y^2 - 6x) = 2x^2y \times 3x^2y^2 - 2x^2y \times 6x $$

Step 2: Multiply the coefficients and add the exponents

First, multiply the coefficients (numbers before the variables) and then add the exponents of the variables that are being multiplied together. $$ 2x^2y \times 3x^2y^2 - 2x^2y \times 6x = (2 \times 3)(x^2 \times x^2)(y \times y^2) - (2 \times 6)(x^2 \times x)(y) $$

Step 3: Simplify the result

Now simplify the multiplication by performing the operations. $$ (6)(x^4)(y^3) - (12)(x^3)(y) = 6x^4y^3 - 12x^3y $$ So, the multiplication and combination of like terms result in the expression: $$ 6x^4y^3 - 12x^3y $$

Key Concepts

Exponents

Exponents play a fundamental role when multiplying polynomials. They are the small numbers located at the top right of a base number or variable, signifying how many times that number is multiplied by itself. For example, in the term \(x^2\), \(x\) is the base, and 2 is the exponent, indicating \(x\) is multiplied by itself twice, resulting in \(x \cdot x\).

When multiplying terms with the same base, you add the exponents together. This is crucial to understand when handling polynomial multiplication. Consider \(x^2 \times x^3\); you add the exponents \(2 + 3\) to get \(x^5\). This rule greatly simplifies working with exponents, especially within polynomial expressions, as demonstrated in the original solution when multiplying terms like \(x^2 \times x^2 = x^4\).

• Add exponents during multiplication if bases match.
• Preserve the base, apply exponents to simplify terms together.

Distributive Property

The distributive property is a core concept in algebra that helps you multiply a single term by a sum or difference within parentheses. It involves 'distributing' the multiplying factor to each term inside the parentheses individually and then performing the multiplication for each pair.

Let's break it down with our example: \(2x^2y(3x^2y^2 - 6x)\). The distributive property allows us to separate this into two distinct multiplications:

• \(2x^2y \times 3x^2y^2\)
• \(- 2x^2y \times 6x\)

Applying this property breaks down complex polynomial expressions into manageable pieces, making it easier to handle each multiplication individually before combining the results. Mastering this property is essential for success in algebra and beyond.

• Used to simplify expressions.
• Helps in reorganizing terms for straightforward multiplications.

Combining Like Terms

Combining like terms is the process of simplifying expressions by merging terms that have identical variable parts. This means that both the variable portion and the exponent must be the same for terms to be combined.

In polynomial multiplication, once you distribute and obtain results, you may encounter terms that are like terms. Although the original solution provided didn't require explicit combining due to the differing exponents, it's helpful to understand that you would sum the coefficients if they had been the same exponent.

• Essential for simplifying expressions after applying the distributive property.
• Helps in reducing complex expressions to simpler forms.

In our specific example, the expression \(6x^4y^3 - 12x^3y\) resulted from distributed multiplication does not contain like terms to combine, as each term has its unique set of exponents, maintaining distinct terms.