Adding and subtracting polynomials is a fundamental skill in algebra. Mastering this allows you to move on to more complex algebraic manipulations and problem-solving. This guide provides a comprehensive walkthrough of the process, complete with examples and practice problems to solidify your understanding.
Understanding Polynomials
Before diving into addition and subtraction, let's ensure we're on the same page regarding polynomials. A polynomial is an expression consisting of variables (usually denoted by x, y, etc.) and coefficients, combined using addition, subtraction, and multiplication, but never division by a variable. Each part of the polynomial separated by a plus or minus sign is called a term. The highest power of the variable in a polynomial is its degree.
Examples:
- 3x² + 5x - 2: This is a polynomial of degree 2 (quadratic). The terms are 3x², 5x, and -2.
- x⁴ - 7x² + 11x: This is a polynomial of degree 4 (quartic).
- 5: This is a polynomial of degree 0 (constant). It's a single term polynomial.
Adding Polynomials
Adding polynomials involves combining like terms. Like terms are terms with the same variable raised to the same power. For instance, 3x² and -2x² are like terms, but 3x² and 5x are not.
Steps:
- Identify like terms: Group the terms with the same variable and exponent together.
- Add the coefficients: Add the numerical coefficients of the like terms.
- Combine: Write the simplified expression.
Example:
Add (4x² + 3x - 7) and (2x² - 5x + 1)
- Identify like terms: (4x² + 2x²) + (3x - 5x) + (-7 + 1)
- Add coefficients: (4 + 2)x² + (3 - 5)x + (-7 + 1)
- Combine: 6x² - 2x - 6
Subtracting Polynomials
Subtracting polynomials is similar to addition, but you must remember to distribute the negative sign to each term in the second polynomial before combining like terms.
Steps:
- Distribute the negative sign: Change the sign of each term in the polynomial being subtracted.
- Identify like terms: Group the like terms together.
- Add the coefficients: Add the coefficients of the like terms.
- Combine: Write the simplified expression.
Example:
Subtract (3x² - 2x + 5) from (7x² + 4x - 1)
This is equivalent to: (7x² + 4x - 1) - (3x² - 2x + 5)
- Distribute the negative sign: 7x² + 4x - 1 - 3x² + 2x - 5
- Identify like terms: (7x² - 3x²) + (4x + 2x) + (-1 - 5)
- Add coefficients: (7 - 3)x² + (4 + 2)x + (-1 - 5)
- Combine: 4x² + 6x - 6
Practice Problems
Try these problems to test your understanding:
- Add (5x³ + 2x² - x + 3) and (x³ - 3x² + 4x - 2).
- Subtract (2x² - 5x + 1) from (8x² + x - 9).
- Add (x⁴ - 2x³ + x) and (3x³ - x² + 5).
- Subtract (y³ - 2y² + 3y - 1) from (4y³ + y² - y + 7).
Remember to show your work step-by-step to ensure accuracy. Mastering these fundamentals will greatly enhance your algebraic skills.
