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Chapter 7: Problem 64

Simplify. $$ a 2-142 a 2-7 a-4-51+2 a $$

Short Answer

Expert verified
-141a^2 - 5a - 55

Step by step solution

01

Group Similar Terms

First, we rearrange the terms of the expression to group similar terms together for simplification. The expression given is \( a^2 - 142a^2 - 7a - 4 - 51 + 2a \). Collect similar terms: \( (a^2 - 142a^2) + (-7a + 2a) + (-4 - 51) \).
02

Simplify Each Group

Now we will simplify each group individually. For the \(a^2\) terms: \(a^2 - 142a^2 = (1 - 142)a^2 = -141a^2\).For the \(a\) terms:\(-7a + 2a = -5a\).For the constants:\(-4 - 51 = -55\).
03

Combine Simplified Groups

Combine all the simplified parts back together to form the fully simplified expression. This results in:\[-141a^2 - 5a - 55\].

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Key Concepts

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

Grouping Similar Terms
When tackling an algebraic expression, the first thing to do is to identify and group similar terms. This is essential because it helps reduce complex expressions into simpler ones, making them easier to manage and solve.
To recognize similar terms, look for those that have the same variable and exponent. For example, in the expression given:
  • a^2 terms are a^2 and -142a^2.
  • a terms are -7a and 2a.
  • Constant terms are -4 and -51.

Once grouped, similar terms can be combined by performing the relevant arithmetic operations (addition or subtraction) on their coefficients. This process is crucial as it sets the stage for further simplifications.
Polynomial Simplification
Polynomial simplification involves breaking down complex polynomial expressions into their simplest form. This is achieved by combining similar terms as previously discussed.
For the expression in hand, once similar terms are grouped, we simplify them one group at a time:
  • For the a^2 terms: Combine them by subtracting coefficients, resulting in -141a^2.
  • For the a terms: Sum the coefficients, giving us -5a.
  • For constant terms: Add them to get -55.

After simplifying each group, polynomials are more easily handled and interpreted, simplifying further calculation and understanding.
Algebraic Subtraction
Algebraic subtraction is the process of subtracting one algebraic expression from another, which frequently occurs when simplifying expressions. In the context of our exercise, it's about reducing the exponents and coefficients within the expression.
To perform algebraic subtraction:
  • Identify terms that can be directly subtracted, such as a^2 - 142a^2.
  • Focus on reducing the coefficients, for instance, 1a^2 - 142a^2 results in -141a^2.
  • Repeat the process for all similar terms, like -7a + 2a leading to -5a.

This method helps in systematically reducing expressions and is a fundamental step in simplifying and solving algebraic equations.

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