6Long Division - dividing a polynomial by a polynomial Think back to long division from 3rd grade:How many times does the divisor go into the dividend? Put that number on top.Multiply that number by the divisor and put the result under the dividend.Subtract and bring down the next number in the dividend. Repeat until you have used all the numbers in the dividend.
10Synthetic Division - To use synthetic division: dividing a polynomial by a polynomialTo use synthetic division:There must be a coefficient for every possible power of the variable.The divisor must have a leading coefficient of 1.
11Step #1: Write the terms of the polynomial so the degrees are in descending order. Since the numerator does not contain all the powers of x, you must include a 0 for the
125 -4 1 6 Since the divisor is x-3, r=3 Step #2: Write the constant r of the divisor x-r to the left and write down the coefficients.5-416Since the divisor is x-3, r=3
1415 15 5 Step #4: Multiply the first coefficient by r, so and place under the second coefficient then add.51515
15Step #5: Repeat process multiplying. the sum, 15, by r; Step #5: Repeat process multiplying the sum, 15, by r; and place this number under the next coefficient, then add.5154541
16Step #5 cont.: Repeat the same procedure. Where did 123 and 372 come from?5154541123372124378
17Step #6: Write the quotient. The numbers along the bottom are coefficients of the power of x in descending order, starting with the power that is one less than that of the dividend.5154541123124372378
18The quotient is:Remember to place the remainder over the divisor.
19Ex 7:Step#1: Powers are all accounted for and in descending order.Step#2: Identify r in the divisor.Since the divisor is x+4, r=-4 .
204 -4 20 8 -5 -1 1 -2 10 Step#3: Bring down the 1st coefficient. Step#4: Multiply and add.Step#5: Repeat.4-4208-5-11-210
21Ex 8:Notice the leading coefficient of the divisor is 2 not 1.We must divide everything by 2 to change the coefficient to a 1.