What should be subtracted from $3 x^{3} - 5 x^{2} + 4 x - 3$ so that the resulting expression has x + 2 as a factor?

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Solution

Let the required number be k.
Let f(x) = $3 x^{3} - 5 x^{2} + 4 x - 3 - k$
According to the given information,
f (-2) = 0
3 ${(- 2)}^{3}$ – 8 ${(- 2)}^{2}$ + 4(-2) – 3 – k = 0
-24 – 32 – 8 – 3 – k = 0
-67 – k = 0
k = -67
Thus, the required number is -67.

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