When \(f(x) = 2x^{3} + mx^{2} + nx + 11\) is divided by \(x^{2} + 5x + 1\), the quotient is \(2x – 5\) and the remainder is \(30x + 16\). Find the values of m and n.
Explanation
To get f(x), multiply the divisor with the quotient and add the remainder.
\(2x^{3} + mx^{2} + nx + 11 = (x^{2} + 5x + 1)(2x - 5) + (30x + 16)\)
= \(2x^{3} - 5x^{2} + 10x^{2} - 25x + 2x - 5 + 30x + 16\)
= \(2x^{3} + 5x^{2} + 7x + 11\)
Therefore, m = 5 and n = 7.