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tony3320 tony3320
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12 years ago
Given that (x^2 - 4) is a factory of the cubic function f(x) = x^3 + cx^2 + dx - 12, find the values of the constants c and d and hence factorise the expression completely.
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NYC23NYC23
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12 years ago
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12 years ago
One way to do this is to recognize that x^2 - 4 = (x + 2)(x - 2).  Do synthetic division with both + and - 2;  each division will end with a remainder in terms of c and d.  Because they are factors, you know that the remainder must be zero.  You will then have two equations to solve for c and d.  See if you can come up with

4c - 2d = 20    and
4c + 2d = 4

Knowing c and d, you can plug them into either of the depressed equations resulting from your synthetic division.  Then you have a simple quadratic to factor:
(x^2 + x - 6)
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