Question

20,21 and question 22

​Q20. Find the values of a it x³ – 5x (a – 1) – 3(x + 1) + 5a is divisible by x – a.

Q21. Find the value of a if x – a is a factor of the polynomial x⁵ – a x⁴ + x³ –a x² + 2x + 3a – 2.

Q22. Find th remainder when x³ + 3px + q is divided by (x² –a²) without actual division.

Answer 1

Dear student
$$\begin{gathered} \mathrm{Que}21 \\ \mathrm{Since}\mathrm{x}-\mathrm{a}\mathrm{is}\mathrm{a}\mathrm{factor}\mathrm{of}{\mathrm{x}}^5-{\mathrm{ax}}^4+{\mathrm{x}}^3-{\mathrm{ax}}^2+2\mathrm{x}+3\mathrm{a}-2=\mathrm{f}\left(\mathrm{x}\right)\left\{\mathrm{let}\right\} \\ \mathrm{so},\mathrm{f}\left(\mathrm{a}\right)=0 \\ \Rightarrow {\mathrm{a}}^5-\mathrm{a}{\left(\mathrm{a}\right)}^4+{\mathrm{a}}^3-\mathrm{a}{\left(\mathrm{a}\right)}^2+2\mathrm{a}+3\mathrm{a}-2=0 \\ \Rightarrow {\mathrm{a}}^5-{\mathrm{a}}^5+{\mathrm{a}}^3-{\mathrm{a}}^3+5\mathrm{a}-2=0 \\ \Rightarrow 5\mathrm{a}-2=0 \\ \Rightarrow 5\mathrm{a}=2 \\ \Rightarrow \mathrm{a}=\frac{2}{5} \end{gathered}$$
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