(3x^3+3x^2-4x+5)+(x^3-2x^2+x-4)

Jasonbradley

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Jun 16, 2024

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Adding Polynomials: A Step-by-Step Guide

This article will guide you through the process of adding two polynomials: (3x^3 + 3x^2 - 4x + 5) + (x^3 - 2x^2 + x - 4).

Understanding Polynomials

A polynomial is an expression consisting of variables and coefficients, combined using addition, subtraction, and multiplication, where the exponents on the variables are non-negative integers.

Key points to remember:

• Terms: A polynomial is made up of terms, separated by addition or subtraction. For example, in the polynomial 3x^3 + 3x^2 - 4x + 5, the terms are 3x^3, 3x^2, -4x, and 5.
• Coefficients: The number in front of a variable is called the coefficient. In the example above, the coefficients are 3, 3, -4, and 5.
• Variables: The letters used in the polynomial are called variables. In this example, the variable is x.
• Exponents: The small number written above and to the right of a variable is the exponent, indicating how many times the variable is multiplied by itself. In our example, the exponents are 3, 2, 1 (implied for x), and 0 (implied for 5).

Adding Polynomials

To add polynomials, we follow these steps:

1. Identify Like Terms: Look for terms with the same variable and the same exponent.
   • Example: In our polynomials, the like terms are:
     • 3x^3 and x^3
     • 3x^2 and -2x^2
     • -4x and x
     • 5 and -4
2. Combine Like Terms: Add the coefficients of like terms while keeping the variable and exponent the same.
   • Example:
     • 3x^3 + x^3 = 4x^3
     • 3x^2 - 2x^2 = x^2
     • -4x + x = -3x
     • 5 - 4 = 1
3. Write the Result: Combine the simplified terms to write the sum of the polynomials.
   • Example: (3x^3 + 3x^2 - 4x + 5) + (x^3 - 2x^2 + x - 4) = 4x^3 + x^2 - 3x + 1

Conclusion

Adding polynomials involves combining like terms by adding their coefficients. By carefully following the steps outlined above, you can successfully add any two polynomials.