1. Dividing Polynomials Using Synthetic Division

2. List all coefficients (numbers in front of x's) and the constant along the top. If a term is missing, put in a 0. 1 Set divisor = 0 and solve. Put answer here. x + 3 = 0 so x = - 3 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). - 31 6 8 -2 1 Bring first number down below line Multiply these and put answer above line in next column - 3 Add these up 3 Multiply these and put answer above line in next column - 9 Add these up - 1 3 1 Multiply these and put answer above line in next column Add these up 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). x 2 + x So the answer is:

3. List all coefficients (numbers in front of x's) and the constant along the top. Don't forget the 0's for missing terms. 1 Set divisor = 0 and solve. Put answer here. x - 4 = 0 so x = 4 Let's try another Synthetic Division 41 0 - 4 0 6 1 Bring first number down below line Multiply these and put answer above line in next column 4 Add these up 4 Multiply these and put answer above line in next column 16 Add these up 12 48 Multiply these and put answer above line in next column Add these up 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 x 3 ). x 3 + x 2 + x + So the answer is: 0 x 3 0 x Multiply these and put answer above line in next column 192 198 Add these up

4. 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 want to divide the factor into the polynomial so set divisor = 0 and solve for first number. Let's try a problem where we factor the polynomial completely given one of its factors. - 24 8 -25 -50 4 Bring first number down below line Multiply these and put answer above line in next column - 8 Add these up 0 Multiply these and put answer above line in next column 0 Add these up - 25 50 0 Multiply these and put answer above line in next column Add these up No remainder so x + 2 IS a factor because it divided in evenly Put variables back in (one x was divided out in process so first number is one less power than original problem). x 2 + x So the answer is the divisor times the quotient: You could check this by multiplying them out and getting original polynomial

5. Acknowledgement I wish to thank Shawna Haider from Salt Lake Community College, Utah USA for her hard work in creating this PowerPoint. www.slcc.edu Shawna has kindly given permission for this resource to be downloaded from www.mathxtc.com and for it to be modified to suit the Western Australian Mathematics Curriculum.www.mathxtc.com Stephen Corcoran Head of Mathematics St Stephen’s School – Carramar www.ststephens.wa.edu.au