The product of two polynomials is $3 x^{4} + 5 x^{3} - 21 x^{2} - 53 x - 30$ . If one of them is $x^{2} - x - 6$ . Find the other polynomial.

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Solution

It is given that one of the polynomial is $x^{2} - x - 6$ and let the other polynomial be $A$ such that the product of these polynomials is $3 x^{4} + 5 x^{3} - 21 x^{2} - 53 x - 30$ that is:
$(x^{2} - x - 6) \times A = 3 x^{4} + 5 x^{3} - 21 x^{2} - 53 x - 30 \Rightarrow A = \frac{3 x^{4} + 5 x^{3} - 21 x^{2} - 53 x - 30}{x^{2} - x - 6}$
Let us now divide $3 x^{4} + 5 x^{3} - 21 x^{2} - 53 x - 30$ by $x^{2} - x - 6$ as shown in the above image:
From the division, we observe that the quotient is $3 x^{2} + 8 x + 5$ and the remainder is $0$ .
Hence, the other polynomial is $3 x^{2} + 8 x + 5$ .

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