Question

find the values of p and q so that x+1 and x-1 are factors of x4+px3+2x2-3x+q

Answer 1

$\textbf{ Hello Mate! }$

What do we mean by factors? Factors means that final value of f(x) will be "0". Hence, value of x = - 1, 1

$f(x) = {x}^{4} + p {x}^{3} + 2 {x}^{2} - 3x + q \\ f( - 1) = { - 1}^{4} + p( { - 1)}^{3} + 2( { - 1)}^{2} - 3( - 1) + q \\ 0 = 1 - p + 2 + 3 + q \\ - 6= - p + q ....(1)\\ \\ f(1) = {1}^{4} + p {(1)}^{3} + 2 {(1)}^{2} - 3(1) + q \\ 0 = 1 + p + 2 - 3 + q \\ 0 = p + q....(2)$

On adding both equations we get

- 6 = - p + q
0 = p + q

- 6 = 2q => q = - 3

Then p = 0 - q => 0 - (-3)
= 3

$\boxed{ \textsf{ Hence,\:p=3\:and\:q=-3 }}$

Have great future ahead!