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Question

# When f(x) = 4x3 − 8x2 + 8x + 1 is divided by a polynomial g(x), we get (2x − 1) as quotient and (x + 3) as remainder. Find g(x).

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Solution

## $\begin{array}{l}\text{Let}f\left(x\right)=\left(4{x}^{3}-8{x}^{2}+8x+1\right),q\left(x\right)=\left(2x-1\right)\text{and}\mathit{\text{r}}\left(x\right)=\left(x+3\right).\\ \text{Then,}f\left(x\right)=g\left(x\right)×q\left(x\right)+r\left(x\right)\\ \\ =>g\left(x\right)=\frac{f\left(x\right)-r\left(x\right)}{q\left(x\right)}\\ \text{Now, [}f\left(x\right)-r\left(x\right)\right]=\left(4{x}^{3}-8{x}^{2}+8x+1\right)-\left(x+3\right)\\ \text{}=\left(4{x}^{3}-8{x}^{2}+7x-2\right)\\ \text{On dividing}\left(4{x}^{3}-8{x}^{2}+7x-2\right)\text{by}\left(2x-1\right),\text{we get}g\left(x\right).\end{array}$ $\begin{array}{l}\\ \therefore g\left(x\right)=2{x}^{2}-3x+2\end{array}$

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