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Chapter 5: Problem 12

Add $$ \left(8 a^{2}+4 a-5\right)+\left(6 a^{2}-3 a-1\right) $$

Short Answer

Expert verified
14a^{2} + a - 6

Step by step solution

01

- Write down the given polynomials

First, clearly write the given polynomials. The expression to be added is equation oindent (8a^{2} + 4a - 5) + (6a^{2} - 3a - 1)
02

- Arrange like terms

Group the like terms together to simplify the addition process. This gives: oindent (8a^{2} + 6a^{2}) + (4a - 3a) + (-5 - 1)
03

- Add the like terms

Add each group of like terms individually. oindent For the quadratic terms: oindent 8a^{2} + 6a^{2} = 14a^{2} oindent For the linear terms: oindent 4a - 3a = a oindent For the constant terms: oindent -5 - 1 = -6
04

- Combine the results

Combine the results from Step 3 to get the final simplified polynomial: oindent 14a^{2} + a - 6

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Key Concepts

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

Polynomial Addition
Adding polynomials means combining them into one polynomial. This is similar to adding numbers, except we deal with variables and their coefficients.
Each term in a polynomial has a specific part it contributes:
  • The variable part, which involves letters like \( a \) or \( x \).
  • The coefficient, which is the number multiplied by the variable.
When adding polynomials, we are essentially adding the coefficients of like terms.
For example, when adding the polynomial \((8a^2 + 4a - 5)\) to \((6a^2 - 3a - 1)\), we write them down, then add each type of term separately.
Like Terms
When working with polynomials, 'like terms' are terms that have the exact same variable part. This means the variables and their exponents should match.
Examples of like terms:
  • \(3a^2 \) and \(5a^2\) (both have \(a^2\))
  • \(-2a \) and \( 7a\) (both have \(a\))
Only like terms can be combined through addition or subtraction. This is why we group them together during polynomial addition.
In the given exercise, we had:
  • Quadratic terms: \(8a^2\) and \(6a^2\)
  • Linear terms: \(4a\) and \(-3a\)
  • Constants: \(-5 \) and \(-1\)
Simplifying Expressions
Simplifying an algebraic expression makes it easier to work with. For polynomial addition, simplifying means combining like terms and ensuring the expression is in its simplest form.
Let's break down this process from the given exercise:
1. First, arrange the like terms together: \( (8a^2 + 6a^2) + (4a - 3a) + (-5 - 1) \) 2. Next, add the like terms individually:
  • Quadratic terms: \(8a^2 + 6a^2 = 14a^2\)
  • Linear terms: \(4a - 3a = a\)
  • Constant terms: \(-5 - 1 = -6\)
3. Finally, combine these results to get the simplified polynomial: \(14a^2 + a - 6\).
Simplifying expressions makes solving equations and further calculations easier.

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