Question

The remainder when the polynomial \( 2x^5 - 3x^4 + 5x^3 - 3x^2 + 7x - 9 \) is divided by \( x^2 - x - 3 \) is:

• A. \( -41x - 3 \)
• B. \( 41x + 3 \)
• C. \( 41x - 3 \)
• D. \( -41x + 3 \)

Hint:

Polynomial Remainders}
When dividing by a degree-2 polynomial, remainder is linear.
Use remainder theorem and evaluate at two roots to form a system.
Alternatively, perform long division.

Answer 1

The Correct Option is B

Let \( P(x) = 2x^5 - 3x^4 + 5x^3 - 3x^2 + 7x - 9 \) We divide \( P(x) \) by \( x^2 - x - 3 \), the remainder will be a linear expression: \[ R(x) = ax + b. \] Use polynomial division or substitution method: Let \( x_1 \) and \( x_2 \) be roots of \( x^2 - x - 3 = 0 \). Plugging those values into \( P(x) \), solve the resulting system of two equations to find \( a \) and \( b \). Doing so gives the remainder: \[ R(x) = 41x + 3. \]