A32 APPENDIX Review To see how synthetic division works, first consider long division for dividing the polynomial − + x x 2 3 3 2 by −x 3. ) + + + − − + − − − x x x x x x x x x x x x 2 5 3 15 2 5 15 3 2 3 6 5 15 15 45 48 3 2 2 3 2 2 2 Check: ⋅ + Divisor Quotient Remainder ( ) ( ) = − + + + = + + − − − + = − + x x x x x x x x x x 3 2 5 15 48 2 5 15 6 15 45 48 2 3 2 3 2 2 3 2 The process of synthetic division arises from rewriting the long division in a more compact form, using simpler notation. For example, in the long division above, the terms in blue are not really necessary because they are identical to the terms directly above them. With these terms removed, we have ) x x x x x x x x x 2 5 15 3 2 3 6 5 15 15 45 48 2 3 2 2 2 + + − − + − − − Most of the x’s that appear in this process can also be removed, provided that we are careful about positioning each coefficient. In this regard, we will need to use 0 as the coefficient of x in the dividend, because that power of x is missing. Now we have ) x x x 5 15 48 2 5 15 3 2 1 0 3 6 15 45 2 + + − − − − − We can make this display more compact by moving the lines up until the numbers in blue align horizontally. ) x x x 2 5 15 3 2 1 0 3 6 15 45 5 15 48 2 + + − − − − − Because the leading coefficient of the divisor is always 1, the leading coefficient of the dividend will also be the leading coefficient of the quotient. So we place the leading coefficient of the quotient, 2, in the circled position. Now, the first three numbers in row 4 are precisely the coefficients of the quotient, and the last number ← Quotient ← Remainder Row 1 Row 2 Row 3 Row 4
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