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Mathematics

For what value of a is the polynomial (2x3 + ax2 + 11x + a + 3) exactly divisible by (2x - 1)?

Factorisation

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Answer

Let f(x) = 2x3 + ax2 + 11x + a + 3.

Given,

Factor : 2x - 1

⇒ 2x - 1 = 0

⇒ 2x = 1

⇒ x = 12\dfrac{1}{2}

Thus, on dividing 2x3 + ax2 + 11x + a + 3 by 2x - 1, remainder = 0.

∴f(12)=0⇒2(12)3+a(12)2+11(12)+a+3=0⇒2(18)+(a4)+(112)+a+3=0⇒(14)+(a4)+(112)+a+3=0⇒(1+a+22+4a+124)=0⇒1+a+22+4a+12=0⇒5a+35=0⇒5a=−35⇒a=−355=−7.\therefore f\Big(\dfrac{1}{2}\Big) = 0 \\[1em] \Rightarrow 2\Big(\dfrac{1}{2}\Big)^3 + a\Big(\dfrac{1}{2}\Big)^2 + 11\Big(\dfrac{1}{2}\Big) + a + 3 = 0 \\[1em] \Rightarrow 2\Big(\dfrac{1}{8}\Big) + \Big(\dfrac{a}{4}\Big) + \Big(\dfrac{11}{2}\Big) + a + 3 = 0 \\[1em] \Rightarrow \Big(\dfrac{1}{4}\Big) + \Big(\dfrac{a}{4}\Big) + \Big(\dfrac{11}{2}\Big) + a + 3 = 0 \\[1em] \Rightarrow \Big(\dfrac{1 + a + 22 + 4a + 12}{4}\Big) = 0 \\[1em] \Rightarrow 1 + a + 22 + 4a + 12 = 0 \\[1em] \Rightarrow 5a + 35 = 0 \\[1em] \Rightarrow 5a = -35 \\[1em] \Rightarrow a = \dfrac{-35}{5} = -7.

Hence, the value of a = -7.

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