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Chapter 10: Problem 13

Subtract the polynomials. $$ (8 a-5)-(3 a+8) $$

Short Answer

Expert verified
The result is \(5a - 13\).

Step by step solution

01

Distribute the Negative Sign

First, distribute the negative sign from the subtraction across the second polynomial. This means you will change the sign of each term inside the parentheses.\[(8a-5)-(3a+8) = 8a - 5 - 3a - 8\]
02

Combine Like Terms

Now, combine the like terms from the expression. This involves adding or subtracting coefficients of the same variable.The like terms here are:- The terms with \(a\): \(8a\) and \(-3a\)- The constant terms: \(-5\) and \(-8\)\[(8a - 3a) + (-5 - 8) = 5a - 13\]

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Key Concepts

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

Distributing the Negative Sign
When subtracting polynomials, distributing the negative sign is crucial. This step changes the operation inside the brackets. The negative sign before a set of parentheses means you need to change the sign of each term within the parentheses.

In our exercise, we have two polynomials to consider:
  • The first polynomial: \(8a - 5\)
  • The second polynomial: \(3a + 8\)
The expression is written as \( (8a - 5) - (3a + 8) \).
To distribute the negative sign, you change the signs in the second polynomial:
  • From \(+3a\) to \(-3a\)
  • From \(+8\) to \(-8\)
After distributing the negative sign, the expression becomes \(8a - 5 - 3a - 8\).

This step simplifies the problem into a straightforward addition problem, where all the terms are directly combined.
Combining Like Terms
After distributing the negative sign, you need to combine like terms to further simplify the expression.

Like terms are terms that have the same variable raised to the same power. In simpler terms, these are terms where the variable and its exponent match.

In the expression \(8a - 5 - 3a - 8\):
  • Identify the terms with the variable \(a\): \(8a\) and \(-3a\).
  • Identify the constant terms: \(-5\) and \(-8\).

To combine the like terms:
  • Add the coefficients of the \(a\) terms: \(8 - 3 = 5\).
  • Add the constant terms: \(-5 - 8 = -13\).
Combining like terms results in \(5a - 13\), giving us a clearer and more direct expression.
Simplifying Expressions
Simplifying expressions means making them as concise as possible, without changing their overall value. In polynomial subtraction, simplifying involves distributing signs properly and carefully combining terms. Once we have our expression \(5a - 13\), it is simplified as much as it can be.

This process strengthens your skills in algebra, providing a basis for more complex operations in mathematics. Simplification ensures expressions are easy to understand and work with.
  • It involves looking for patterns and reducing expressions to their simplest forms.
  • This practice makes calculations more feasible and less error-prone.
Every step taken in simplifying an expression contributes to grasping the essence of the problem-solving process in algebra, making you more adept and confident in handling other mathematical challenges.

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