Polynomials are the building blocks of algebra. They are expressions made up of variables, coefficients, and exponents. One of the most fundamental operations you'll perform with them is addition and subtraction. This process involves identifying and combining "like terms" to simplify a longer expression into its most compact form. Our Adding and Subtracting Polynomials Calculator is designed to help students master this core skill, providing a quick way to check homework and visualize how complex expressions can be neatly simplified.
How to Use the Polynomials Calculator
Simplifying your polynomial expression is easy with our tool:
- Enter the First Polynomial: Input the coefficients for each term (x³, x², x, and the constant) of your first polynomial. Use '0' for any missing terms.
- Select the Operator: Choose whether you want to add (+) or subtract (-) the two polynomials.
- Enter the Second Polynomial: Input the coefficients for your second polynomial.
- Calculate the Result: Click the "Calculate" button to see the simplified polynomial expression.
The Key Concept: Combining Like Terms
The entire process of adding and subtracting polynomials hinges on one simple idea: you can only combine terms that are "alike."
What Are Like Terms?
Like terms are terms that have the exact same variable part—meaning the same variable(s) raised to the exact same power. The numerical coefficients in front of them can be different.
- Like Terms:
7x²and-2x²(both have x²) - Like Terms:
4yandy(both have y) - Not Like Terms:
3xand3x²(the exponents are different)
Think of it like sorting fruit. You can add three apples and four apples to get seven apples. But you can't add three apples and four oranges to get "seven apple-oranges." In the same way, you can only add or subtract the coefficients of terms that are of the same kind.
The Process of Combining
- Identify Like Terms: Group together all the terms with the same variable and exponent.
- Add or Subtract Coefficients: For each group of like terms, simply add or subtract their numerical coefficients. The variable part stays the same.
An Example of Addition
Let's add (2x² + 6x + 5) and (3x² - 2x - 1).
- Combine the x² terms: 2x² + 3x² = 5x²
- Combine the x terms: 6x - 2x = 4x
- Combine the constants: 5 - 1 = 4
The result is 5x² + 4x + 4.
An Example of Subtraction
Subtraction has one extra, crucial step: you must first distribute the negative sign to *every term* in the second polynomial you are subtracting.
Let's subtract (5x² - 3x + 2) from (7x² + 5x + 6). The problem is (7x² + 5x + 6) - (5x² - 3x + 2).
- Distribute the negative: The expression becomes (7x² + 5x + 6) - 5x² + 3x - 2.
- Combine the x² terms: 7x² - 5x² = 2x²
- Combine the x terms: 5x + 3x = 8x
- Combine the constants: 6 - 2 = 4
The result is 2x² + 8x + 4. Forgetting to distribute the negative is the most common mistake students make.
Polynomial Terminology
- Term: A single part of a polynomial, which can be a number, a variable, or a product of numbers and variables (e.g., 5x²).
- Coefficient: The number in front of the variable (e.g., in 5x², the coefficient is 5).
- Degree of a Term: The exponent of the variable in that term.
- Degree of a Polynomial: The highest degree of any of its terms.
- Standard Form: Writing a polynomial by putting the terms in order from highest degree to lowest degree. Our calculator presents the results in standard form.
Frequently Asked Questions
What if a term is missing in one of the polynomials?
If a term is missing, its coefficient is simply zero. Our calculator handles this automatically if you enter '0' or leave the field blank for that term. For example, the polynomial 2x² + 5 is the same as 2x² + 0x + 5.
Does the order matter when adding polynomials?
No. Addition is commutative, which means A + B is the same as B + A. You will get the same result regardless of which polynomial you enter as Polynomial 1 or Polynomial 2.
Does the order matter when subtracting polynomials?
Yes, absolutely. Subtraction is not commutative. (A - B) is not the same as (B - A). You must be careful to enter the polynomial you are subtracting from as Polynomial 1 and the one being subtracted as Polynomial 2.