(x^5+3x^4-x+7)+(3x^5-8x^4-x^3+12)

Jasonbradley

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Jun 17, 2024

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Simplifying Polynomials: A Step-by-Step Guide

This article will guide you through the process of simplifying the expression: (x^5+3x^4-x+7)+(3x^5-8x^4-x^3+12)

Understanding the Process

The expression involves polynomials, which are algebraic expressions made up of variables and constants combined using addition, subtraction, multiplication, and non-negative integer exponents.

To simplify this expression, we'll use the following steps:

1. Identify like terms: Like terms are terms with the same variable and exponent. For example, '3x^4' and '-8x^4' are like terms.

2. Combine like terms: Add or subtract the coefficients of like terms.

Step-by-Step Solution

Let's break down the simplification:

1. (x^5+3x^4-x+7)+(3x^5-8x^4-x^3+12)

2. Identify like terms:

   • x^5 terms: x^5 and 3x^5
   • x^4 terms: 3x^4 and -8x^4
   • x^3 terms: -x^3
   • x terms: -x
   • Constant terms: 7 and 12

3. Combine like terms:

   • x^5 terms: x^5 + 3x^5 = 4x^5
   • x^4 terms: 3x^4 - 8x^4 = -5x^4
   • x^3 terms: -x^3
   • x terms: -x
   • Constant terms: 7 + 12 = 19

4. Combine all terms:

   • 4x^5 - 5x^4 - x^3 - x + 19

Final Answer

The simplified form of the expression (x^5+3x^4-x+7)+(3x^5-8x^4-x^3+12) is 4x^5 - 5x^4 - x^3 - x + 19.