The polynomial \(2x^{3} + 3x^{2} + qx - 1\) has the same reminder when divided

Explanation

Using the remainder theorem, the remainder when a polynomial \(ax^{2} + bx + c\) is divided by \((x - a)\) is equal to \(f(a)\).

\(2x^{3} + 3x^{2} + qx - 1\) divided by \((x + 2)\), the remainder = \(f(-2)\)

\(\implies f(-2) = f(1)\)

\(f(-2) = 2(-2^{3}) + 3(-2^{2}) + q(-2) - 1 = -16 + 12 - 2q - 1 = -5 - 2q\)

\(f(1) = 2(1^{3}) + 3(1^{2}) + q(1) - 1 = 2 + 3 + q - 1 = 4 + q\)

\(4 + q = -5 -2q \implies 4 + 5 = -2q - q = -3q\)

\(q = -3\)