Addition and Subtraction of Polynomials Calculator

Enter an expression with the polynomials to add or subtract. You can enter multiple expressions and use grouping symbols (parentheses, brackets, braces).

Quick Examples

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How to Use the Calculator

This online polynomial addition and subtraction calculator is an algebraic tool designed to simplify mathematical expressions. In addition to providing the final simplified result, the solver generates a step-by-step solution so you can understand how signs are handled and terms are combined.

How to enter your data:

  • Enter the complete expression: use the single text field to input the entire operation at once. Use parentheses to separate each polynomial, for example: (3x2-2x+5) - (x2+4x-1) + (2x-3). You can include monomials, binomials, trinomials, or polynomials of any size.
  • Use any numerical format: the calculator supports integer, rational (exact fractions), and irrational coefficients (such as roots of any index). Enter them exactly as they appear in your problem; the solver will keep them in symbolic form to avoid rounding errors.

Once you enter the expression, the math engine will show you a detailed step-by-step solution following this order:

  1. Evaluate previous operations: if your exercise includes pending powers or multiplications (for example, a scalar multiplying a polynomial like 3(x2+2)), the solver will respect the order of operations and solve them before continuing.
  2. Remove parentheses: the step-by-step solution will show you how grouping symbols are removed by applying the distributive property. If a parenthesis is preceded by a negative sign, the algorithm will automatically change the sign of all terms inside it.
  3. Combine like terms: with the expression free of parentheses, the calculator will classify the elements. It will add and subtract only those terms that share the same variable and exponent.
  4. Final simplified result: as the last step, you will get the fully simplified polynomial. By mathematical convention, it will be presented in descending order by exponent, ending with the constant term if the expression has one.

Solved Exercises

The following are examples of problems solved by the calculator.

Calculate this polynomial addition: \(\left(3x^3+5x^2-2x\right)+\left(2x^3-4x^2+7\right).\)

Result

$$ 5 x^{3} + x^{2} - 2 x + 7 $$

Step-by-step solution

1. Analyze the original expression.

We start with the entered expression:

$$ \left(3x^3+5x^2-2x\right)+\left(2x^3-4x^2+7\right) $$

2. Remove the parentheses.

Since these are additions, the signs of all terms remain unchanged:

$$ 3 x^{3} + 5 x^{2} - 2 x + 2 x^{3} - 4 x^{2} + 7 $$

3. Group like terms.

We rearrange the expression by placing like terms next to each other in descending order based on their degree to make the operation easier:

$$ 3 x^{3} + 2 x^{3} + 5 x^{2} - 4 x^{2} - 2 x + 7 $$

4. Combine like terms.

Combine the terms of degree 3:

$$ 5 x^{3} + 5 x^{2} - 4 x^{2} - 2 x + 7 $$

Combine the terms of degree 2:

$$ 5 x^{3} + x^{2} - 2 x + 7 $$
Find the result of this polynomial subtraction: \(\left(x^2+2x\right)-\left(x-7+x^2\right).\)

Result

$$ x + 7 $$

Step-by-step solution

1. Analyze the original expression.

We start with the entered expression:

$$ \left(x^2+2x\right)-\left(x-7+x^2\right) $$

2. Remove the parentheses.

We change the signs of all terms inside the polynomial preceded by a negative sign:

$$ x^{2} + 2 x - x + 7 - x^{2} $$

3. Group like terms.

We rearrange the expression by placing like terms next to each other in descending order based on their degree to make the operation easier:

$$ x^{2} - x^{2} + 2 x - x + 7 $$

4. Combine like terms.

Combine the terms of degree 2:

$$ 2 x - x + 7 $$

Combine the terms of degree 1:

$$ x + 7 $$
Determine the result of the subtraction between the polynomials \(\left(4x^4-2x^2+x\right)\) and \(\left(x^4+3x^2-5\right).\)

Result

$$ 3 x^{4} - 5 x^{2} + x + 5 $$

Step-by-step solution

1. Analyze the original expression.

We start with the entered expression:

$$ \left(4x^4-2x^2+x\right)-\left(x^4+3x^2-5\right) $$

2. Remove the parentheses.

We change the signs of all terms inside the polynomial preceded by a negative sign:

$$ 4 x^{4} - 2 x^{2} + x - x^{4} - 3 x^{2} + 5 $$

3. Group like terms.

We rearrange the expression by placing like terms next to each other in descending order based on their degree to make the operation easier:

$$ 4 x^{4} - x^{4} - 2 x^{2} - 3 x^{2} + x + 5 $$

4. Combine like terms.

Combine the terms of degree 4:

$$ 3 x^{4} - 2 x^{2} - 3 x^{2} + x + 5 $$

Combine the terms of degree 2:

$$ 3 x^{4} - 5 x^{2} + x + 5 $$
Calculate the following subtraction of binomials: \((2x^3+5x^2)-(3x^2-1).\)

Result

$$ 2 x^{3} + 2 x^{2} + 1 $$

Step-by-step solution

1. Analyze the original expression.

We start with the entered expression:

$$ (2x^3+5x^2)-(3x^2-1) $$

2. Remove the parentheses.

We change the signs of all terms inside the polynomial preceded by a negative sign:

$$ 2 x^{3} + 5 x^{2} - 3 x^{2} + 1 $$

3. Combine like terms.

Combine the terms of degree 2:

$$ 2 x^{3} + 2 x^{2} + 1 $$

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Daniel Machado

Professor of Mathematics, graduated from the Faculty of Exact, Chemical and Natural Sciences of the National University of Misiones (UNAM). Developer and creator of RigelUp, dedicated to building tools for mathematical learning.

Last updated: (5 days ago)