Finding a Polynomial Using Division Algorithm

Video Explanation

Question

The polynomial which when divided by

\[ -x^2 + x – 1 \]

gives quotient

\[ x – 2 \]

and remainder

\[ 3 \]

is:

(a) \(x^3 – 3x^2 + 3x – 5\)
(b) \(-x^3 – 3x^2 – 3x – 5\)
(c) \(-x^3 + 3x^2 – 3x + 5\)
(d) \(x^3 – 3x^2 – 3x + 5\)

Solution

Step 1: Use the Division Algorithm

By the division algorithm,

\[ \text{Dividend} = (\text{Divisor}) \times (\text{Quotient}) + \text{Remainder} \]

Step 2: Substitute the Given Values

\[ f(x) = (-x^2 + x – 1)(x – 2) + 3 \]

Step 3: Multiply

\[ (-x^2 + x – 1)(x – 2) \]

\[ = -x^3 + 2x^2 + x^2 – 2x – x + 2 \]

\[ = -x^3 + 3x^2 – 3x + 2 \]

Step 4: Add the Remainder

\[ f(x) = -x^3 + 3x^2 – 3x + 2 + 3 \]

\[ = -x^3 + 3x^2 – 3x + 5 \]

Conclusion

The required polynomial is:

\[ \boxed{-x^3 + 3x^2 – 3x + 5} \]

Hence, the correct option is (c).