%2B(8x%5E4-3x%5E3%2B4x%2B1).jpeg "(12x^5-3x^4+2x-5)+(8x^4-3x^3+4x+1)")
Adding Polynomials: A Step-by-Step Guide
In this article, we'll explore how to add two polynomials: (12x⁵ - 3x⁴ + 2x - 5) + (8x⁴ - 3x³ + 4x + 1). Let's break down the process step by step.
Understanding Polynomials
Polynomials are expressions that consist of variables and constants combined using addition, subtraction, multiplication, and non-negative integer exponents. Each term in a polynomial is a product of a coefficient (a constant) and a variable raised to a power.
Adding Polynomials
To add polynomials, we simply combine like terms. Like terms have the same variable raised to the same power.
Step 1: Identify Like Terms
- x⁵ terms: 12x⁵
- x⁴ terms: -3x⁴ and 8x⁴
- x³ terms: -3x³
- x terms: 2x and 4x
- Constant terms: -5 and 1
Step 2: Combine Like Terms
Add the coefficients of each set of like terms:
- 12x⁵
- (-3x⁴ + 8x⁴) = 5x⁴
- -3x³
- (2x + 4x) = 6x
- (-5 + 1) = -4
Step 3: Write the Sum
Combine the simplified terms to get the final result:
(12x⁵ - 3x⁴ + 2x - 5) + (8x⁴ - 3x³ + 4x + 1) = 12x⁵ + 5x⁴ - 3x³ + 6x - 4
Conclusion
Adding polynomials involves identifying like terms, combining their coefficients, and writing the sum in standard form. This process allows us to simplify expressions and perform operations on polynomials effectively.