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Chapter 4: Problem 66

Perform each indicated operation. \(\left(x^{2}+x\right)-\left(3 x^{2}+2 x-1\right)\)

Short Answer

Expert verified
-2x^{2} - x + 1

Step by step solution

01

Distribute the Negative Sign

Rewrite the expression making sure to distribute the negative sign across the terms in the second parenthesis. \[(x^{2} + x) - (3x^{2} + 2x - 1) \rightarrow x^{2} + x - 3x^{2} - 2x + 1\]
02

Combine Like Terms

Combine the like terms (terms involving the same power of x) to simplify the expression. Terms involving \(x^{2}\): \(x^{2} - 3x^{2}\) Terms involving \(x\): \(x - 2x\) Constant terms: \(+1\)
03

Simplify Each Group

Simplify each group from Step 2. \( x^{2} - 3x^{2} = -2x^{2} \) \( x - 2x = -x \) The constant term remains 1.
04

Write the Final Answer

Combine all the simplified terms from Step 3 to give the final answer: \[ -2x^{2} - x + 1 \]

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Key Concepts

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

Distributing Negative Signs
When you see a subtraction sign in front of a parenthesis, it means you need to distribute the negative sign across all the terms inside the parenthesis. This can be expressed as multiplying each term by -1. Distributing removes the parenthesis and changes the signs of the terms inside.
For example, in the exercise \((x^{2} + x) - (3x^{2} + 2x - 1)\), distribute the negative sign to get: \((x^{2} + x) - 3x^{2} - 2x + 1\).
Breaking that down:
  • \text{The term } 3x^{2} \text{ becomes } -3x^{2}.
  • \text{The term } 2x \text{ becomes } -2x.
  • \text{The term } -1 \text{ becomes } +1.
So, you'll get: \[x^{2} + x - 3x^{2} - 2x + 1 \].
Combining Like Terms
After distributing any negative signs, the next step is to combine like terms. Like terms are terms that have the same variables raised to the same power.
  • \text{In our expression } (x^{2} + x - 3x^{2} - 2x + 1), \text{ the terms with } x^{2} \text{ are } x^{2} \text{ and } -3x^{2}.
  • \text{The terms with } x \text{ are } x \text{ and } -2x.
  • \text{The constant term is } +1 \text{ as it has no variable.}
Combine these by simply adding or subtracting the coefficients:
  • \text{For } x^{2}: x^{2} - 3x^{2} = -2x^{2}
  • \text{For } x: x - 2x = -x
  • \text{Constant term stays } +1
You now have the expression: \( -2x^{2} - x + 1 \).
Simplifying Expressions
Simplifying expressions involves putting the expression in its most compact and readable form.
After distributing the negative signs and combining like terms, what remains is often already the simplest form. Here, we move towards expressing the equation as clearly as possible.
  • \text{Ensure all like terms are combined (as done in the previous step)}.
  • \text{Reorder the terms (if needed) in a standard form where exponents are in descending order: } -2x^{2} \text{ is written first, followed by terms in } x, \text{ and then constants}.
So for our example: \( -2x^{2} - x + 1 \). This is the most simplified form.
By following these steps: distributing negative signs, combining like terms, and simplifying the expression, you can handle polynomial operations effectively.

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