Quick Math Polynomial Division- Division

Quick Math Polynomial Division

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

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