What must be added to $5 x^{3} - 2 x^{2} + 6 x + 7$ to make the sum $x^{3} + 3 x^{2} - x + 1$ ?

4 x^{3} - 5 x^{2} + 7 x + 6

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4 x^{3} + 5 x^{2} - 7 x + 6

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- 4 x^{3} + 5 x^{2} - 7 x - 6

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Solution

The correct option is C $- 4 x^{3} + 5 x^{2} - 7 x - 6$
$(x^{3} + 3 x^{2} - x + 1) - (5 x^{3} - 2 x^{2} + 6 x + 7)$
$= (1 - 5) x^{3} + (3 + 2) x^{2} - (1 + 6) x + 1 - 7$
$= - 4 x^{3} + 5 x^{2} - 7 x - 6$
Hence $(- 4 x^{3} + 5 x^{2} - 7 x - 6)$ must be added to $(x^{3} + 3 x^{2} - x + 1)$ to make the sum $(5 x^{3} - 2 x^{2} + 6 x + 7)$ .

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