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Chapter 6: Problem 42

Simplify. $$ \left(4 x^{2}-3 y^{2}+5 x y\right)-\left(8 x y+3 y^{2}\right) $$

Short Answer

Expert verified
The simplified expression is \(4x^2 - 6y^2 - 3xy\).

Step by step solution

01

Distribute the Negative Sign

The problem is to simplify the expression \( (4x^2 - 3y^2 + 5xy) - (8xy + 3y^2) \). We start by distributing the negative sign to the terms inside the second parenthesis. This gives us \( 4x^2 - 3y^2 + 5xy - 8xy - 3y^2 \).
02

Combine Like Terms

Now, we combine like terms. Start with the \( y^2 \) terms: \(-3y^2 - 3y^2\) which simplifies to \(-6y^2\). Next, combine the \( xy \) terms: \(5xy - 8xy\) is \(-3xy\). The \( x^2 \) term \(4x^2\) remains unchanged. Thus, the expression simplified is \(4x^2 - 6y^2 - 3xy\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Distributive Property
The distributive property is an essential algebraic concept used to simplify expressions. It allows us to remove parentheses by distributing a factor across terms inside the parentheses. In the expression \( (4x^2 - 3y^2 + 5xy) - (8xy + 3y^2) \), the negative sign outside the second set of parentheses is distributed to each term inside, effectively changing the sign of each term. Here's how it works:
  • The negative sign multiplies each term inside \(- (8xy + 3y^2)\).
  • This results in \(-8xy - 3y^2\).
Combining these, the expression becomes:\( 4x^2 - 3y^2 + 5xy - 8xy - 3y^2 \).Understanding the distributive property helps in algebraic manipulations, ensuring expressions are simplified accurately.
Like Terms
Like terms are terms in an algebraic expression that have identical variable parts raised to the same power. They can be combined by adding or subtracting their coefficients. This is key to simplifying expressions.In the expression \( 4x^2 - 3y^2 + 5xy - 8xy - 3y^2 \), like terms include:
  • The \( y^2 \) terms: \(-3y^2\) and \(-3y^2\) which combine to \(-6y^2\).
  • The \( xy \) terms: \( 5xy \) and \(-8xy\) which combine to \(-3xy\).
  • The \( x^2 \) term \(4x^2\) which stands alone.
Understanding how to combine like terms simplifies complex algebraic expressions and is crucial for mathematical problem-solving.
Algebraic Expressions
An algebraic expression is a mathematical phrase that can contain numbers, variables, and operators (such as +, -, *, and /). In algebraic expressions, terms are separated by addition or subtraction.The original expression\( (4x^2 - 3y^2 + 5xy) - (8xy + 3y^2) \)is an example of a situation where simplifying is needed to remove unnecessary complexity. Simplification involves:
  • Using the distributive property to eliminate parentheses, as explained previously.
  • Identifying and combining like terms to achieve a simpler form.
The result of simplifying the given expression is a more straightforward version,\( 4x^2 - 6y^2 - 3xy \), that makes it easier to understand and use in further mathematical calculations. Understanding these components is vital for mastering algebra.

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Most popular questions from this chapter

For Exerises \(26-31\) , complete each of the following. a. Graph each funnction by making a table of values. b. Determine the consecutive integer values of \(x\) between which each real zero is located. C. Estimate the \(x\) -coordinates at which the relative maxima and relative minima occur. $$ f(x)=x^{4}-9 x^{3}+25 x^{2}-24 x+6 $$

PERSONAL FINANCE For Exercises \(38-41,\) use the following information. Zach has purchased some home theater equipment for \(\$ 2000,\) which he is financing through the store. He plans to pay \(\$ 340\) per month and wants to have the balance paid off after six months. The formula \(B(x)=2000 x^{6}-\) 340\(\left(x^{5}+x^{4}+x^{3}+x^{2}+x+1\right)\) represents his balance after six months if \(x\) represents 1 plus the monthly interest rate (expressed as a decimal). Find his balance after 6 months if the annual interest rate is 12\(\% .\) (Hint: The monthly interest rate is the annual rate divided by \(12,\) so \(x=1.01 . )\)

For Exercises \(39-41,\) sketch a graph of each polynomial. odd-degree polynomial function with one relative maximum and one relative minimum; the leading coefficient is negative

Given a polynomial and one of its factors, find the remaining factors of the polynomial. Some factors may not be binomials. $$ 2 x^{3}-5 x^{2}-28 x+15 ; x-5 $$

Find the greatest common factor of each set of numbers. $$ 24,84 $$
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