Problem 99 - Elementary Algebra

Question

For the following problems, perform the multiplications and combine any like terms. $$ x^{3} y^{2}\left(5 x^{2} y^{2}-3\right)(2 x y-1) $$

Step-by-Step Solution

Answer

Question: Multiply the given expressions and simplify the resulting expression: $$x^{3} y^{2}(5 x^{2} y^{2}-3)(2 x y-1)$$ Answer: $$10 x^{8} y^{7} - 6 x^{7} y^{5} - 5 x^{8} y^{6} + 3 x^{3} y^{2}$$

Step 1: Expand the expression

To expand the given expression, we will apply the distributive property by multiplying each term in the second parentheses with each term in the first parentheses: $$ x^{3} y^{2}(5 x^{2} y^{2}-3)(2 x y-1) \Rightarrow x^{3} y^{2}( 5 x^{2} y^{2}(2 x y) - 3(2 x y) + 5 x^{2} y^2(-1) -3(-1) ) $$

Step 2: Multiply the terms

Next, we will multiply the terms inside the expression to simplify it further: $$ x^{3} y^{2}( 10 x^{5} y^{5} - 6 x^{4} y^3 - 5 x^{5} y^{4} + 3 ) $$

Step 3: Combine like terms

Finally, we will combine any like terms present in the simplified expression. In this case, there are no terms with the same power of x and y, so the resulting expression is: $$ 10 x^{8} y^{7} - 6 x^{7} y^{5} - 5 x^{8} y^{6} + 3 x^{3} y^{2} $$

Key Concepts

Distributive Property

Understanding the distributive property is crucial in simplifying algebraic expressions, particularly when expanding polynomials. The distributive property allows one to multiply a term efficiently across a group of terms inside a parenthesis. For the problem presented, think of distributing as handing out the term outside the parentheses to each term inside the parentheses one by one. This action ensures that every part of the expression gets multiplied, leading to a proper expansion.

The expression given in the problem, \(x^{3} y^{2}(5x^{2} y^{2}-3)(2xy-1)\), uses the distributive property by taking \(x^{3} y^{2}\) and applying it to each element in both \( (5x^{2} y^{2}-3)\) and \((2xy-1)\). By applying the distributive property:

• First multiply \(5x^{2} y^{2} imes 2xy\) and \(5x^{2} y^2 imes (-1)\).
• Then multiply \(-3 imes 2xy\) and \(-3 imes (-1)\).

This method will help expand and simplify the expression, making it easier to manage and understand.

Like Terms

In polynomials and algebraic expressions, combining like terms is an essential simplifying step. Like terms are terms in an expression that have identical variable parts, including the same powers. Only coefficients of these terms differ and can be combined together.

Let's unpack this with an example. Consider the expression from the problem: \(10 x^{8} y^{7} - 6 x^{7} y^{5} - 5 x^{8} y^{6} + 3 x^{3} y^{2}\). Although no like terms were found in the final expression, it's good practice to examine expressions closely. Like terms were initially checked after performing all multiplications and expansions. In this simplified polynomial:

• \(10 x^{8} y^{7}\), \(-6 x^{7} y^{5}\), \(-5 x^{8} y^{6}\), and \(3 x^{3} y^{2}\) are unique due to differing variable powers.

If applicable, like terms are added or subtracted together, streamlining the final expression.
This step helps reduce complexity and prepares the polynomial for any further algebraic operations or evaluations.

Algebraic Expressions

Understanding algebraic expressions is foundational for mastering polynomials and higher-level mathematics. An algebraic expression consists of variables, constants, and arithmetic operations. It can involve one or multiple terms connected by addition or subtraction.

The problem at hand starts with an algebraic expression: \(x^{3} y^{2}(5x^{2} y^{2}-3)(2xy-1)\).

• Variables: Letters such as \(x\) and \(y\) that represent unknown values.
• Constants: Numbers like 3 and 5 that remain unchanged.
• Operations: Includes addition, subtraction, and multiplication.

In simplifying such expressions, one deals with addition, subtraction, multiplication, and occasionally division as well.
Grasping the concept of algebraic expressions equips you with the tools to approach more complex problems while clearly communicating how quantities relate to one another.