Question

Solve this:
17. In each of the following two polynomials, find the value of a, if x - a is a factor:
(i) x⁶ - ax⁵ + x⁴ - ax³ + 3x - a + 2 (ii) x⁵ - a²x³ - 2x + a + 1

Answer 1

Dear student
$$\begin{gathered} \mathrm{Let}\mathrm{f}\left(\mathrm{x}\right)={\mathrm{x}}^6-{\mathrm{ax}}^5+{\mathrm{x}}^4-{\mathrm{ax}}^3+3\mathrm{x}-\mathrm{a}+2\mathrm{be}\mathrm{the}\mathrm{given}\mathrm{polynomial}. \\ \mathrm{By}\mathrm{the}\mathrm{factor}\mathrm{theorem},\left(\mathrm{x}-\mathrm{a}\right)\mathrm{is}\mathrm{the}\mathrm{factor}\mathrm{of}\mathrm{f}\left(\mathrm{x}\right),\mathrm{if}\mathrm{f}\left(\mathrm{a}\right)=0\mathrm{i}.\mathrm{e}., \\ \mathrm{f}\left(\mathrm{a}\right)={\mathrm{a}}^6-\mathrm{a}{\left(\mathrm{a}\right)}^5+{\mathrm{a}}^4-\mathrm{a}{\left(\mathrm{a}\right)}^3+3\mathrm{a}-\mathrm{a}+2=0 \\ {\mathrm{a}}^6-{\mathrm{a}}^6+{\mathrm{a}}^4-{\mathrm{a}}^4+3\mathrm{a}-\mathrm{a}+2=0 \\ 2\mathrm{a}=-2 \\ \mathrm{a}=-1 \\ \mathrm{Thus},\mathrm{the}\mathrm{value}\mathrm{of}\mathrm{a}\mathrm{is}-1. \end{gathered}$$
Similarly try other part.
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