As an alternative to evaluating polynomials by direct substitution, nested factoring can be used. The method has the advantage of using only products and sums−no powers. For P = x3 + 3x2+ 1x+ 5, we begin by grouping all variable terms and factoring x: P = [x3 + 3x2 + 1x] + 5 = x[x2 + 3x + 1] + 5. Then we group the inner terms with x and factor again: P = x[x2 + 3x+ 1] + 5 = x[x(x+ 3) + 1] + 5.
The expression can now be evaluated using any input and the order of operations. If x= 2, we quickly find that P = 27. Use this method to evaluate H = x3 + 2x2 + 5x− 9 for x= −3.
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