Dividing Polynomials © 2002 by Shawna Haider

3x3 4x2 x 2 1 1 1 1 Dividing by a Monomial If the divisor only has one term, split the polynomial up into a fraction for each term. divisor Now reduce each fraction. 3x3 4x2 x 2 1 1 1 1

Subtract (which changes the sign of each term in the polynomial) Long Division If the divisor has more than one term, perform long division. You do the same steps with polynomial division as with integers. Let's do two problems, one with integers you know how to do and one with polynomials and copy the steps. Subtract (which changes the sign of each term in the polynomial) Now multiply by the divisor and put the answer below. x + 11 2 1 Bring down the next number or term Multiply and put below Now divide 3 into 5 or x into 11x First divide 3 into 6 or x into x2 32 698 x - 3 x2 + 8x - 5 Remainder added here over divisor 64 x2 – 3x subtract 5 8 11x - 5 32 11x - 33 This is the remainder 26 28 So we found the answer to the problem x2 + 8x – 5  x – 3 or the problem written another way:

Remainder added here over divisor Let's Try Another One If any powers of terms are missing you should write them in with zeros in front to keep all of your columns straight. Subtract (which changes the sign of each term in the polynomial) y - 2 Write out with long division including 0y for missing term Multiply and put below Bring down the next term Multiply and put below y + 2 y2 + 0y + 8 Divide y into y2 Divide y into -2y Remainder added here over divisor y2 + 2y subtract -2y + 8 - 2y - 4 This is the remainder 12

- 3 1 6 8 -2 - 3 - 9 3 1 x2 + x 3 - 1 1 1 Synthetic Division There is a shortcut for long division as long as the divisor is x – k where k is some number. (Can't have any powers on x). Set divisor = 0 and solve. Put answer here. x + 3 = 0 so x = - 3 1 - 3 1 6 8 -2 Multiply these and put answer above line in next column Multiply these and put answer above line in next column Multiply these and put answer above line in next column Bring first number down below line - 3 Add these up - 9 3 Add these up Add these up 1 x2 + x 3 - 1 1 This is the remainder Put variables back in (one x was divided out in process so first number is one less power than original problem). So the answer is: List all coefficients (numbers in front of x's) and the constant along the top. If a term is missing, put in a 0.

Let's try another Synthetic Division 0 x3 0 x Set divisor = 0 and solve. Put answer here. x - 4 = 0 so x = 4 1 4 1 0 - 4 0 6 Multiply these and put answer above line in next column Multiply these and put answer above line in next column Multiply these and put answer above line in next column Multiply these and put answer above line in next column Bring first number down below line 4 Add these up 16 48 192 Add these up Add these up Add these up 1 x3 + x2 + x + 4 12 48 198 This is the remainder Now put variables back in (remember one x was divided out in process so first number is one less power than original problem so x3). So the answer is: List all coefficients (numbers in front of x's) and the constant along the top. Don't forget the 0's for missing terms.

Let's try a problem where we factor the polynomial completely given one of its factors. You want to divide the factor into the polynomial so set divisor = 0 and solve for first number. - 2 4 8 -25 -50 Multiply these and put answer above line in next column Multiply these and put answer above line in next column Multiply these and put answer above line in next column Bring first number down below line - 8 Add these up 50 Add these up Add these up No remainder so x + 2 IS a factor because it divided in evenly 4 x2 + x - 25 Put variables back in (one x was divided out in process so first number is one less power than original problem). So the answer is the divisor times the quotient: List all coefficients (numbers in front of x's) and the constant along the top. If a term is missing, put in a 0. You could check this by multiplying them out and getting original polynomial