Question

How is polynomial long division like long division with whole numbers? How is it different?

Short answer

Polynomial long division and long division of whole numbers both operate on the principle of repeated subtraction. The process for both involves dividing, multiplying, and subtracting until the remainder is less than the divisor. The main difference is in what we are dividing. In whole number division, we divide simple numbers whereas in polynomial long division, we divide terms of polynomials, focusing on the highest degree terms.

Step-by-step solution

Understand Long Division with Whole Numbers

Standard long division with whole numbers involves repeatedly subtracting the divisor from the dividend, each time recording the result until the remaining dividend is less than the divisor. In each step we multiply the divisor by an integer, subtract it from the dividend and write down the result. We then bring down a digit from the original dividend and repeat the process.

Understand Polynomial Long Division

In polynomial long division, the process is quite similar. The main difference lies in what we are dividing. Instead of whole numbers, we are using polynomials. The process involves dividing the highest degree term of the dividend by the highest degree term of the divisor and writing the result on the quotient line. We then multiply the entire divisor by this quotient, subtract it from the dividend and write down the result. This new polynomial becomes the dividend for the next step, and the process is repeated until the degree of the divisor is greater than the degree of the dividend.

Compare and Contrast

Both the divisions works in the same fundamental principle: division as repeated subtraction. The main difference lies in the nature of the numbers being divided (whole numbers vs polynomials). Moreover, when dividing polynomials, attention must be paid to the degree of the terms. The divisor must always divide the term with the highest degree in the dividend.