Dividing a Polynomial by a Binomial The process of dividing a polynomial by a binomial (or any polynomial other than a monomial) is referred to as long division. The process is very similar to performing long division with positive integers.

Example No.3 Divide (x2 + 7x + 12) by (x + 3). Solution Begin by writing the problem in “long division” form. Make sure that the divisor and dividend are written in descending order and that any “missing” terms are accounted for with a zero. (This is not an issue with this example.)

Example No.3 Solution (continued) Now divide the leading term in the divisor into the leading term of the dividend. The result is the first part of the quotient. Multiply the result of this division by the divisor by using the Distributive Law. Then subtract this from the dividend. You have to remember that you are subtracting! Using parentheses and the subtraction sign usually helps.

Example No.3 Solution (continued) Now “bring down” all of the remaing parts of the divided. Now divide the leading term in the divisor into the leading term of the current remainder, which is 4x + 12. The result is the next part of the quotient. Multiply 4 by the divisor and subtract from the current remainder. Since the remainder is 0, the division is complete.

Check for Example No.3

Example No.4 Divide (15x2  22x + 14) by (3x  2). Solution

Example No.4 Solution (continued) We can express the result of this division in two ways: (1) The answer is 5x  4 with R6 (2)

Check for Example No.4 (divisor)(quotient) + remainder = dividend

Example No.5 Divide (x5  3x4  4x2 + 10x) by (x  3). Solution Note that there are “missing” terms in the dividend. It is best if you account for these by using 0 as a “place-holder.”

Example No.5 Solution (continued) The answer is

Check for Example No.5

Reminders for Polynomial Division Write the dividend in descending order. Account for any missing terms in the dividend by writing the missing term with a coefficient of 0. Remember that you are subtracting at each step – be sure to distribute the negative through before subtracting. Check your work with multiplication.