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Chapter 2: Problem 20

Simplify. $$ 6 x 2-4 x+7 x 2-3 x $$

Short Answer

Expert verified
The simplified expression is \(13x^2 - 7x\).

Step by step solution

01

Identify Like Terms

In the expression \(6x^2 - 4x + 7x^2 - 3x\), identify terms with the same variables and exponents. Here, \(6x^2\) and \(7x^2\) are like terms, and \(-4x\) and \(-3x\) are like terms.
02

Combine Like Terms for \(x^2\)

Add the like terms \(6x^2\) and \(7x^2\) to get a single term. This gives \((6 + 7)x^2 = 13x^2\).
03

Combine Like Terms for \(x\)

Add the like terms \(-4x\) and \(-3x\) to get a single term. This gives \((-4 - 3)x = -7x\).
04

Write the Simplified Expression

Combine the results from Steps 2 and 3 to get the simplified expression: \(13x^2 - 7x\).

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Key Concepts

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

Understanding Like Terms
Like terms are an essential concept in simplifying polynomials. They are terms that contain the exact same variables raised to the same powers.
For example, in the expression \(6x^2 - 4x + 7x^2 - 3x\), \(6x^2\) and \(7x^2\) are like terms because they both involve \(x\) raised to the power of 2. Similarly, \(-4x\) and \(-3x\) are like terms as they both involve \(x\) to the power of 1.
It's crucial to only combine terms that are like terms, as combining unlike terms can lead to incorrect simplification.
  • Identify terms with the same base and exponent as like terms.
  • Remember that coefficients can be different; only the variables and their exponents must match.
How to Combine Terms
Once like terms have been identified, the next step is to combine them. Combining terms involves adding or subtracting their coefficients.
For our example expression \(6x^2 + 7x^2\), we add the coefficients (6 and 7) to get \(13x^2\). The variable part \(x^2\) remains unchanged because we only focus on the coefficients.
This step helps reduce the expression to fewer terms, making it simpler and easier to understand.
  • Add the coefficients of like terms.
  • Keep the variable part the same; don鈥檛 change its power.
  • Apply the same principle to negative terms: for \(-4x - 3x\), combine to get \(-7x\).
Breaking Down Algebraic Expressions
Algebraic expressions are combinations of numbers and variables connected by basic operations like addition and subtraction.
Understanding algebraic expressions is crucial because it forms the foundation of algebra, enabling you to simplify, solve, and manipulate equations.
In the expression \(6x^2 - 4x + 7x^2 - 3x\), we see several parts:
  • Terms, which are parts separated by plus or minus signs, like \(6x^2\), \(-4x\), \(7x^2\), and \(-3x\).
  • Coefficients, which are the numerical parts in front of the variables (e.g., 6, -4, 7, -3).
  • Variables, which are symbols that represent numbers (\(x\) in this case).
Recognizing each component of an algebraic expression helps in simplifying it and solving related problems more effectively.

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