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Subtract The Following Polynomials Calculator

Reviewed by Calculator Editorial Team

This calculator helps you subtract polynomials with variables and exponents. Polynomials are algebraic expressions consisting of variables and coefficients, combined using addition, subtraction, multiplication, and non-negative integer exponents.

How to Subtract Polynomials

Subtracting polynomials follows the same rules as adding polynomials, but with the operation of subtraction. Here's a step-by-step guide:

  1. Write both polynomials in standard form, ordered from the highest degree to the lowest.
  2. Align like terms (terms with the same variable and exponent) vertically.
  3. Subtract the coefficients of like terms.
  4. Combine the remaining terms to form the result.

Remember that when subtracting terms with the same variable and exponent, you subtract their coefficients. If a term exists in one polynomial but not the other, it remains in the result with its original sign.

Polynomial Subtraction Formula

The general formula for subtracting two polynomials is:

(a₁xⁿ + a₂xⁿ⁻¹ + ... + aₙx + a₀) - (b₁xᵐ + b₂xᵐ⁻¹ + ... + bₘx + b₀) = (a₁xⁿ - b₁xᵐ) + (a₂xⁿ⁻¹ - b₂xᵐ⁻¹) + ... + (a₀ - b₀)

Where:

  • a₁, a₂, ..., a₀ are the coefficients of the first polynomial
  • b₁, b₂, ..., b₀ are the coefficients of the second polynomial
  • n and m are the highest exponents in each polynomial

Note: For subtraction, the coefficients of the second polynomial are subtracted from the coefficients of the first polynomial for like terms.

Polynomial Subtraction Examples

Example 1: Simple Polynomial Subtraction

Subtract 3x² + 2x + 5 from 5x² + 4x + 1:

(5x² + 4x + 1) - (3x² + 2x + 5) = (5x² - 3x²) + (4x - 2x) + (1 - 5) = 2x² + 2x - 4

Example 2: Subtracting with Missing Terms

Subtract x³ + 2x from 3x³ + 5x² + x:

(3x³ + 5x² + x) - (x³ + 2x) = (3x³ - x³) + 5x² + (x - 2x) = 2x³ + 5x² - x

Example 3: Subtracting with Negative Coefficients

Subtract -2x² + 3x - 7 from 4x² - x + 2:

(4x² - x + 2) - (-2x² + 3x - 7) = (4x² + 2x²) + (-x - 3x) + (2 + 7) = 6x² - 4x + 9

Polynomial Subtraction FAQ

What is the difference between subtracting polynomials and adding polynomials?
The main difference is the operation performed on the coefficients of like terms. When adding, you add the coefficients, while when subtracting, you subtract the coefficients of the second polynomial from the first.
Can I subtract polynomials with different numbers of terms?
Yes, you can. Terms that exist in only one polynomial will remain in the result with their original sign. For example, subtracting x² + x from 3x³ + 2x would result in 3x³ + x² - x.
What happens if I subtract a polynomial from itself?
Subtracting a polynomial from itself will result in zero, assuming the polynomials are identical. For example, (2x² + 3x + 1) - (2x² + 3x + 1) = 0.
Is polynomial subtraction commutative?
No, polynomial subtraction is not commutative. This means that A - B does not equal B - A. For example, (3x + 2) - (x + 1) = 2x + 1, while (x + 1) - (3x + 2) = -2x - 1.