oo 11m (l+l/x)x by tabulating values for x = 1, 10, 100, 1000, ...

J[I]Ol 1. J[I]OOl 1. 19CJClCA9 -0. 1999 -0. 2. RULES FOR DIFFERENTIATION The simple theorems we saw in the last chapter about sums, products, and quotients of limits readily yield THEOREMS ABOUT DERIVATIVES: If f and g are functions that have derivatives at a, and b is a reaZ number, then (f+g) '(a) (bf) , (a) = f' (a) + g' (a) bf' (a) (fg) , (a) = f' (a)g(a) + f(a)g' (a) (f/g) , (a) = g(a)f' (a) - f(a)g' (a) g(a)2 40 where the last rule, for quotients, only makes sense when g(a) ~ O. Also, we see from the definition of derivative that if f(x) = b is a constant function, then f'(a) = 0 for every number a.

Hence qo ~ 1, ql = Z2 and q2 = Z2 2 3, while r O. Thus 3 synthetic division shows that x 3 - with zero for remainder. 5320&89. The first of these values is the zero 23 for f(x) that we have already found above. 532D88R. Hence synthetic division offers an alternative method of finding the remaining zeros of a cubic polynomial once one zero is known. EXAMPLE: 4x 3 + 3x 2 - 2x - 1 = 0 We rewrite this equation as x = 1/(4x 2 2x 1 2) 1 + 3x - 2). This defines the algorithm which we begin blindly with a guess of Xo = O.