Inserting literal factors, a^ -+2 a^b + 3 a^b^ + ab^ is the complete quotient, and — 5 a^b^ + 3 a^b + ab^ is the remainder.
Explanation. The term ab^ in the divisor is missing, so we write for the coefficient of this term in the column of figures on the left of the vertical line. We add the columns as in Ex. 1, but as the first term of the divisor is 2, we divide each sum by 2 before placing the result in the line of quotients. We then use these quotients as multipliers, the multiplicand being in each case — 3, 0, and 1, and form the horizontal lines as in Ex. 1. Having obtained the required number of terms in the quotient, the remainder is found by adding the rest of the columns and setting down the results vjithotit dividing by 2. By continuing the first horizontal line (dividend), as shown in this example, we at once see what literal factors the remainder must contain.
EXAMPLES V. d.
Divide:
1. a^ - 4 a^ + 2 a^ +4a + 1 by a^ - 2 a - 1. 2. a^ - 4 a^b + 6 a^b^ + b^4 -4ab^3 by a^2 + b^2 - 2 ab. 3. a5 - 10 a^b + 16 a^b^ -12ab^ + ab4 + 2 b^5 by (a- b)^. 4. x^ - 2 b^x^ + b^8 by x^ + bx^ + b^x + b^. 5. x^ - 3 x^y^ + 8xy^ — 5y^ by x^ -4xy -- y^ to four terms in the quotient.
