Question

For what value of a is 2x³+ax²+11x+a+3 exactly divisible by 2x-1

Answer 1

hey mate here is ur ans
see the pic
hope it is not wrong and helps you

Answer 2

Answer:

Let

$\boxed{ \sf p(x) = 2 {x}^{3} + ax {}^{2} + 11x + a + 3}$

be the given polynomial. If p(x) is exactly divisible by 2x - 1, then (2x-1) is a factor of p(x).

$\\ \therefore \sf p \bigg( \frac{1}{2} \bigg) = 0 \qquad \bigg \{ \because2x - 1 = 0 \rightarrow \: x = \frac{1}{2} \bigg \} \\ \\ \\ \implies \sf2 \times \bigg( \frac{1}{2} \bigg) {}^{3} + a \bigg( \frac{1}{2} \bigg) {}^{2} + 11 \times \frac{1}{2} + a + 3 = 0 \\ \\ \\ \implies \sf \frac{1}{4} + \frac{a}{4} + \frac{11}{2} + a + 3 = 0 \\ \\ \\ \implies \sf \frac{1 + a + 22 + 4a + 12}{4 } = 0 \\ \\ \\ \implies \sf \frac{5a + 35}{4} = 0 \\ \\ \\ \implies \sf5a + 35 = 0 \\ \\ \\ \implies \sf \blue{a = - 7}$

Thus, the given polynomial is divisible by 2x-1, if a = -7.