Polynomial Long Division

Polynomial long division is a method for dividing one polynomial (the dividend) by another (the divisor), which can be a monomial, binomial, or any other polynomial. Similar to numerical division, this method yields a quotient and a remainder.

When written as a fraction, the polynomial in the numerator is the dividend, and the polynomial in the denominator is the divisor.

Steps

Now, let us divide 3x² – 5x + 2 by x – 1 using the long-division method.

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

Arranging Terms of the Dividend 

First, we will arrange the terms in the decreasing order of their indices (if required) and write the missing terms of the dividend with a coefficient of 0. 

Here, the numerator 3x² – 5x + 2 and the denominator x – 1 are already arranged in descending order of their indices.

Thus, the dividend is 3x² – 5x + 2

The divisor is x – 1

Dividing the First Term of the Dividend

On dividing the first term of the dividend 3x² by the first term of the divisor x, we get

${\dfrac{3x^{2}}{x}=3x}$

Thus, the first term of the quotient is 3x

Multiplying the Result and the Divisor 

On multiplying the entire divisor, x – 1, by the result 3x, we get the product 

3x(x – 1) = 3x² – 3x

Subtracting the Product From the Dividend

Now, we subtract the product, 3x² – 3x, from the dividend, 3x² – 5x + 2, and bring down the next term, 2. The difference, -2x + 2, along with the brought-down term, 2, will form the new dividend.

On subtracting the product from the dividend and bringing down the next term (i.e., +2), we get the new dividend, which is -2x + 2

Repeating the Steps (Until the Remainder is of Lower Degree)

Now, we will repeat the process until the remainder is either 0 or of a lower degree than the divisor and can no longer be divided.

On repeating the same process, we get

Thus, the quotient is 3x – 2, and the remainder is 0

With Missing Terms

Often, a polynomial expression may have some missing term (for example, there may be a x³, but no x²). In such a case, we either leave gaps or write the missing terms with a coefficient of 0. 

Let us divide the 4^th-degree polynomial 6x⁴ – 5x + 1 by the binomial x² + 1

${\dfrac{6x^{4}-5x+1}{x^{2}+1}}$

Here, the quotient is 6x² – 6, and remainder is -5x + 7