Question

If quadratic polynomial ax2 + bx + c is such that when it is divided by x, (x -1) and (x + 1) the remainders are 3, 6, 4 respectively, then the value of a + b is

Answer 1

Dear student
$$\begin{gathered} \mathrm{Since}{\mathrm{ax}}^2+\mathrm{bx}+\mathrm{c}\mathrm{when}\mathrm{divided}\mathrm{by}\mathrm{x},\left(\mathrm{x}-1\right),\left(\mathrm{x}+1\right)\mathrm{leaves}\mathrm{the}\mathrm{remainder}3,6\mathrm{and}4 \\ \mathrm{respectively}.\mathrm{So}\mathrm{by}\mathrm{division}\mathrm{algorithm},\mathrm{we}\mathrm{get} \\ {\mathrm{ax}}^2+\mathrm{bx}+\mathrm{c}=\left(\mathrm{mx}+\mathrm{n}\right)\times \mathrm{x}+3 \\ \mathrm{On}\mathrm{comparing}\mathrm{we}\mathrm{get}, \\ \mathrm{a}=\mathrm{m},\mathrm{b}=\mathrm{n}\mathrm{and}\mathrm{c}=3...\left(1\right) \\ \mathrm{Again}, \\ {\mathrm{ax}}^2+\mathrm{bx}+\mathrm{c}=\left(\mathrm{px}+\mathrm{q}\right)\times \left(\mathrm{x}-1\right)+6 \\ {\mathrm{ax}}^2+\mathrm{bx}+\mathrm{c}={\mathrm{px}}^2-\mathrm{px}+\mathrm{qx}-\mathrm{q}+6 \\ {\mathrm{ax}}^2+\mathrm{bx}+\mathrm{c}={\mathrm{px}}^2+\mathrm{x}\left(\mathrm{q}-\mathrm{p}\right)+6-\mathrm{q} \\ \mathrm{On}\mathrm{comparing}\mathrm{we}\mathrm{get}, \\ \mathrm{a}=\mathrm{p},\mathrm{b}=\mathrm{q}-\mathrm{p}\mathrm{and}\mathrm{c}=6-\mathrm{q}\Rightarrow \mathrm{q}=3\left[\mathrm{from}1\right] \\ \mathrm{and}\mathrm{b}=3-\mathrm{p}\Rightarrow \mathrm{b}=3-\mathrm{a}\Rightarrow \mathrm{a}+\mathrm{b}=3 \end{gathered}$$
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