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Remainder and Factor Theorems

Question

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Using remainder theorem, find the value of k if on dividing $2 x^{3} + 3 x^{2} - k x + 5 by x - 2$ , leaves a remainder 7.

Solution

Let f ( x ) = 2 x³ + 3 x² - k x + 5

Using remainder theorem,

F ( 2 ) = 7

$∴ 2 (2)^{3} + 3 (2)^{2} - k (2) + 5 = 7 ∴ 2 (8) + 3 (4) - k (2) + 5 = 7 ∴ 16 + 12 - 2 k + 5 = 7 ∴ 2 k = 16 + 12 + 5 - 7 ∴ 2 k = 26 ∴ k = 13$

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