2. Dividing by a Monomial If the divisor only has one term, split the polynomial up into a fraction for each term. Now reduce each fraction. 3 x 3 4 x 2 x 2 1 1 1 1 divisor
3. 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. x - 3 x 2 + 8 x - 5 2 x 64 x 2 – 3 x Subtract (which changes the sign of each term in the polynomial) 5 11 x 8 - 5 1 + 11 Multiply and put below 32 11 x - 33 subtract 26 28 This is the remainder Remainder added here over divisor First divide 3 into 6 or x into x 2 Now divide 3 into 5 or x into 11 x 32 698 Now multiply by the divisor and put the answer below. Bring down the next number or term So we found the answer to the problem x 2 + 8 x – 5 x – 3 or the problem written another way:
4. 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. y + 2 y 2 + 0 y + 8 y y 2 + 2 y Subtract (which changes the sign of each term in the polynomial) -2 y + 8 - 2 Multiply and put below - 2 y - 4 subtract 12 This is the remainder Remainder added here over divisor Bring down the next term Multiply and put below Divide y into -2 y Divide y into y 2 Write out with long division including 0 y for missing term
5. 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 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 ). - 3 1 6 8 -2 1 - 3 Add these up 3 - 9 Add these up - 1 3 1 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: Set divisor = 0 and solve. Put answer here. x + 3 = 0 so x = - 3 Bring first number down below line 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
6. 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 Let's try another Synthetic Division 4 1 0 - 4 0 6 1 4 Add these up 4 16 Add these up 12 48 48 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 192 198 Add these up Set divisor = 0 and solve. Put answer here. x - 4 = 0 so x = 4 Bring first number down below line 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
7. 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. - 2 4 8 -25 -50 4 - 8 Add these up 0 0 Add these up - 25 50 0 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 Bring first number down below line 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
