What must be added to $2 x^{4} - 5 x^{3} + 2 x^{2} - x - 3$ so that the result is exactly divisible by (x- 2)?

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Solution

ANSWER:
Let k be added to $2 x^{4} - 5 x^{3} + 2 x^{2} - x - 3$ so that the result is exactly divisible by ( x – 2). Here, k is a constant.
∴ $f (x) = 2 x^{4} - 5 x^{3} + 2 x^{2} - x - 3 + k$ is exactly divisible by ( x – 2).
Using factor theorem, we have
f(2)=0

⇒ $2 \times 2^{4} - 5 \times 2^{3} + 2 \times 2^{2} - 2 - 3 + k = 0 \Rightarrow 32 - 40 + 8 - 5 + k = 0 \Rightarrow - 5 + k = 0 \Rightarrow k = 5$
Thus, 5 must be added to $2 x^{4} - 5 x^{3} + 2 x^{2} - x - 3$ so that the result is exactly divisible by ( x – 2).

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