If x - 2 is a factor of each of the following two polynomials, find the value of a in each case:
1. $x^{3}-2 a x^{2}+a x-1$
2. $x^{5}-3 x^{4}-a x^{3}+3 a x^{2}+2 a x+4$
(1) Let $f(x)=x^{3}-2 a x^{2}+a x-1$
from factor theorem
if (x - 2) is the factor of f(x) the f(2) = 0
let, x - 2 = 0
⟹ x = 2
Substitute x value in f(x)
$f(2)=2^{3}-2 a(2)^{2}+a(2)-1$
= 8 - 8a + 2a - 1
= - 6a + 7
Equate f(2) to zero
⟹ - 6a + 7 = 0
⟹ - 6a = -7
⟹ a= 76
When, (x - 2) is the factor of f(x) then a = 76
(2) Let, $f(x)=x^{5}-3 x^{4}-a x^{3}+3 a x^{2}+2 a x+4$
from factor theorem
if (x - 2) is the factor of f(x) the f(2) = 0
let, x - 2 = 0
⟹ x = 2
Substitute x value in f(x)
$f(2)=2^{5}-3(2)^{4}-a(2)^{3}+3 a(2)^{2}+2 a(2)+4$
= 32 - 48 - 8a + 12 + 4a + 4
= 8a - 12
Equate f(2) to zero
⟹ 8a - 12 = 0
⟹ 8a = 12
⟹ a = 12/8
= 3/2
So, when (x - 2) is a factor of f(x) then a = 32
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