Question

If 2x + 1 is a factor of (3b + 2)x³ + (b $-$ 1) then find b.

Answer 1

$$\begin{gathered} \mathrm{Let}p\left(x\right)=\left(3b+2\right)x^3+\left(b-1\right) \\ \mathrm{Given}:2x+1\mathrm{is}\mathrm{a}\mathrm{factor}\mathrm{of}p\left(x\right). \\ \therefore \mathrm{When}\mathrm{we}\mathrm{divide}p\left(x\right)\mathrm{by}2x+1,\mathrm{the}\mathrm{remainder}\mathrm{will}\mathrm{be}\mathrm{zero}. \\ \mathrm{Here},\mathrm{divisor}=2x+1 \\ \mathrm{When}2x+1=0,\mathrm{we}\mathrm{have}x=-\frac{1}{2} \\ \therefore \mathrm{Remainder}=p\left(-\frac{1}{2}\right)=0 \\ \Rightarrow \left(3b+2\right){\left(-\frac{1}{2}\right)}^3+\left(b-1\right)=0 \\ \Rightarrow \left(3b+2\right)\left(-\frac{1}{8}\right)+\left(b-1\right)=0 \\ \Rightarrow \frac{-\left(3b+2\right)+8\left(b-1\right)}{8}=0 \\ \Rightarrow -\left(3b+2\right)+8\left(b-1\right)=0 \\ \Rightarrow -3b-2+8b-8=0 \\ \Rightarrow 5b-10=0 \\ \Rightarrow 5b=10 \\ \therefore b=2 \end{gathered}$$