Question

If 3x – 1 is a factor of the polynomial 81x³ – 45x² +3a – 6, then the value of a is ___________.

Answer 1

Let f(x) = 81x³ – 45x² +3a – 6

It is given that one factor of f(x) is (3x – 1).

Therefore, $f\left(x\right)=0\mathrm{when}x=\frac{1}{3}$.

On putting x = $\frac{1}{3}$ in f(x) = 0, we get

$$\begin{gathered} 81{\left(\frac{1}{3}\right)}^3-45{\left(\frac{1}{3}\right)}^2+3a-6=0 \\ \Rightarrow 81\left(\frac{1}{27}\right)-45\left(\frac{1}{9}\right)+3a-6=0 \\ \Rightarrow 3-5+3a-6=0 \\ \Rightarrow 3a-8=0 \\ \Rightarrow 3a=8 \\ \Rightarrow a=\frac{8}{3} \end{gathered}$$

Hence, if 3x – 1 is a factor of the polynomial 81x³ – 45x² +3a – 6, then the value of a is $\underline{\frac{8}{3}}.$