Subtraction of Polynomials

Definition

The subtraction of polynomials involves subtracting corresponding terms from each polynomial. Corresponding terms are those that have the same variable raised to the same power. The subtraction is done by subtracting the coefficients of these corresponding terms while keeping the variable part unchanged.

Steps for Subtracting Polynomials

1. Arrange Polynomials: Write each polynomial in a standard form (i.e., in descending order of the degrees of its terms).
2. Identify Like Terms: Identify the terms in both polynomials that have the same variables raised to the same powers.
3. Subtract the Coefficients: Subtract the coefficients of the like terms in the second polynomial from the corresponding coefficients in the first polynomial.
4. Write the Result: Combine the results to form the new polynomial.

Examples

Example 1: Subtracting Two Polynomials

Subtract the polynomial \(x^2 + 3x + 4\) from \(3x^2 + 5x + 6\).

First Polynomial: \(x^2 + 3x + 4\)

Second Polynomial: \(3x^2 + 5x + 6\)

The difference is:

Example 2: Subtracting Polynomials with Different Degrees

Subtract the polynomial \(x^3 + 4x^2 - 2x + 1\) from \(2x^3 + 3x^2 + x + 5\).

First Polynomial: \(x^3 + 4x^2 - 2x + 1\)

Second Polynomial: \(2x^3 + 3x^2 + x + 5\)

The difference is:

Example 3: Subtracting Polynomials with Missing Terms

Subtract the polynomial \(4x^3 + x^2 - 3\) from \(6x^3 + 2x - 7\).

First Polynomial: \(4x^3 + x^2 - 3\)

Second Polynomial: \(6x^3 + 2x - 7\)

The difference is:

Example 4: Subtracting Polynomials with Negative Coefficients

Subtract the polynomial \(-3x^2 + 2x - 5\) from \(-x^2 - 4x + 1\).

First Polynomial: \(-3x^2 + 2x - 5\)

Second Polynomial: \(-x^2 - 4x + 1\)

The difference is: