Question

Please help me with 51 question

Answer 1

Steps:
1) If (x-1) &(2x-1) are the factors of some polynomial

$p(x) = p {x}^{2} + 5x + r$

Then,
We can say :

$p(x) = a(x -2)(2x - 1) \\ = > p(x) = a(2 {x}^{2} - 4x - x + 2)$
where a is some constant.

$p {x}^{2} + 5x + r= a(2 {x}^{2} - 5x + 2)$

Comparing coefficient of x, we get
a = -1
=> p = r = -2 
Hence proved.