## Quick Math Polynomial Division

##### Enter expression in this form

Example: 2x^3 + 3x^2 - 5x + 6 , x - 1

The dividend and divisor are separated by comma.

### Polynomial Division of Algebraic Expressions

Polynomial division is a process similar to long division with numbers, but it involves polynomials. When dividing polynomials, we typically use the long division method or synthetic division (if the divisor is a binomial of the form \( x – c \)). Here, we’ll focus on the long division method.

#### Steps for Polynomial Long Division:

1. Arrange: Write both the dividend and the divisor in descending order of their degrees.

2. Divide: Divide the first term of the dividend by the first term of the divisor to get the first term of the quotient.

3. Multiply: Multiply the entire divisor by the first term of the quotient.

4. Subtract: Subtract the result from the dividend to form a new polynomial.

5. Repeat: Repeat the process using the new polynomial as the dividend until the degree of the remainder is less than the degree of the divisor.

#### Detailed Example:

Let’s divide \( 2x^3 + 3x^2 – 5x + 6 \) by \( x – 1 \).

1. Arrange the polynomials:

– Dividend: \( 2x^3 + 3x^2 – 5x + 6 \)

– Divisor: \( x – 1 \)

2. Divide the first term:

– \( \frac{2x^3}{x} = 2x^2 \)

– Quotient: \( 2x^2 \)

3. Multiply:

– \( 2x^2 \cdot (x – 1) = 2x^3 – 2x^2 \)

4. Subtract:

\[

(2x^3 + 3x^2 – 5x + 6) – (2x^3 – 2x^2) = 3x^2 – (-2x^2) – 5x + 6 = 5x^2 – 5x + 6

\]

5. Repeat the process with the new dividend \( 5x^2 – 5x + 6 \):

– \( \frac{5x^2}{x} = 5x \)

– Quotient: \( 2x^2 + 5x \)

– Multiply: \( 5x \cdot (x – 1) = 5x^2 – 5x \)

– Subtract:

\[

(5x^2 – 5x + 6) – (5x^2 – 5x) = 6

\]

6. Final Step:

– New dividend: \( 6 \)

– Since the degree of the remainder (0) is less than the degree of the divisor (1), the division process stops.

The final quotient is \( 2x^2 + 5x + 1 \) and the remainder is \( 7 \).

So,

\[ \frac{2x^3 + 3x^2 – 5x + 6}{x – 1} = 2x^2 + 5x + 1 + \frac{7}{x – 1} \]