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Chapter 4: Problem 81

Add or subtract as indicated. \((4 x+2 x y-3)-(-2 x+3 x y+4)\)

Short Answer

Expert verified
6x - xy - 7

Step by step solution

01

- Distribute Negative Sign

First, distribute the negative sign into each term inside the parentheses of the second polynomial: -( -2 x + 3 x y + 4 ). This becomes 2x - 3xy - 4.
02

- Rewrite the Expression

Rewrite the expression without the parentheses: (4x + 2xy - 3) + ( 2x - 3xy - 4 )
03

- Combine Like Terms

Combine like terms: 4x and 2x, 2xy and -3xy, and -3 and -4.This results in: (4x + 2x) + (2xy - 3xy) + (-3 - 4).
04

- Simplify

Simplify each set of like terms: 6x - xy - 7.

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Key Concepts

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

Distributing Negative Signs
Understanding how to distribute negative signs is crucial when working with polynomials. When you see a negative sign in front of parentheses, it means you need to multiply every term inside the parentheses by -1. This changes the sign of each term inside the parentheses:
For example, if you have -( -2x + 3xy + 4), each term inside needs its sign flipped:
  • -(-2x) becomes 2x
  • -(3xy) becomes -3xy
  • -(4) becomes -4
So, the expression -( -2x + 3xy + 4) turns into 2x - 3xy - 4.
Being careful with signs is essential to avoid mistakes in polynomial operations.
Combining Like Terms
Combining like terms helps to simplify polynomials by grouping similar types of terms together. Like terms are terms that have the same variable raised to the same power.
For example, in the expression (4x + 2xy - 3) + ( 2x - 3xy - 4), we identify the like terms:
  • 4x and 2x
  • 2xy and -3xy
  • -3 and -4
Once we've grouped like terms, we combine them by adding or subtracting their coefficients:
  • 4x + 2x = 6x
  • 2xy - 3xy = -xy
  • -3 - 4 = -7
Thus, our combined expression is 6x - xy - 7. Combining like terms ensures our polynomial is in its simplest form.
Simplifying Expressions
Simplifying expressions is all about making an equation as straightforward as possible. After combining like terms, always double-check if you can simplify further. For our expression 6x - xy - 7, there are no more like terms to combine, so it's already simplified.
Simplifying often involves:
  • Combining like terms
  • Distributing any remaining constants
  • Removing parentheses when possible
Remember, a simplified expression is typically easier to work with and understand. It helps to always re-check your work to ensure accuracy in your results.

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Most popular questions from this chapter

Graph each equation by completing the table of values. $$ \begin{aligned} &y=-x^{2}+4\\\ &\begin{array}{c|c} \hline x & y \\ \hline-2 & \\ \hline-1 & \\ \hline 0 & \\ \hline 1 & \\ \hline 2 & \end{array} \end{aligned} $$

Add or subtract as indicated. \((-5 t+13 s)+(8 t-3 s)\)

Light travels at a speed of \(1.86 \times 10^{5} \mathrm{mi}\) per sec. When Venus is \(6.68 \times 10^{7} \mathrm{mi}\) from the sun, how long (in seconds) does it take light to travel from the sun to Venus? (Data from The World Almanac and Book of Facts.)

If one "dog" year is estimated to be about seven "human" years, the monomial \(7 x\) gives the dog's age in human years for \(x\) dog years. Evaluate this monomial for \(x=9\). Use the result to fill in the blanks: If a dog is ______ in dog years, then it is _________ in human years.

Without actually performing the operations, mentally determine the coefficient of the \(x^{2}\) -term in the simplified form of $$\left(-4 x^{2}+2 x-3\right)-\left(-2 x^{2}+x-1\right)+\left(-8 x^{2}+3 x-4\right)$$
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