Vol. 4 No 1 - Pi Mu Epsilon

We now illustrate this rule with some exam les.Let us find thequotient and the remainder when f(x) = 2x3 - 3x5 - 3x + 5 is divided by2x - 1. First we divide f(x) by x - 1/2 thus:On dividing the quotient 2x 2 - 2x - 4 by 2, we find that the requiredquotient is x 2 - x - 2 and the remainder is 3. Let us now find thequotient and remainder when f (x) = 2x3 - 3x 2 - 3x + 5 is divided byx/2 + 1. First we divide £(x by x + 2 thus:Sow, dividing the quotient 2x' - 7x + 11 by 1/2, we find that the requiredquotient is 4x 2 - 14x + 22 and the remainder is -17.From these equations it is easily seen that each coefficient after thesecond in the quotient is formed by multiplying the two precedingcoefficients by b and c respectively and adding these products to thenext coefficient in the dividend. The process of finding the coefficientsin the quotient and remainder can be arranged as follows:This process can easily be extended to divisors of degree higherthan the first. The extension will now be made only to divisors of thesecond degree, since it is similar for divisors of higher degree.Suppose that a polynomialis divided by the trinomial x 2 - bx - c.Letbe the quotient and let R = px + q be the remainder.thatf (XI = (x2 - bx - c)Q(x) + R,n-2+ b Xn-4 + -*-3 4= (x2 - bx - c) (bx + b x ~ - ~Then it followsExpanding the right-hand member of equation (4) and equating coefficientsof like powers of x, we find thatExplanation: First, place the last two coefficients of the divisor withsigns changed on the left of the vertical line. Add the first columnon the right of the vertical line. This gives a the first numberbelow the horizontal line. Next, multiply b andOcOiy%i and write theseproducts in the second row and in the second and third columns. Next,add the second column. This gives a + bb2 or b Multiply b and c by3"b3 and write these products in the third row and in the third and fourthcolumns. Next, add the third column. This gives a2 + bb3 + cb2 or b4.Continue this process until an entry is made in the last column.add the last two columns to find p and q.Then