Question # If ( p(x)=x^{3}-3 x^{2}+2 x+5 ) and ( p(a)=p(b)=p(c)=0, ) then the value of ( (2-a)(2-b)(2-c) ) is

# If ( p(x)=x^{3}-3 x^{2}+2 x+5 ) and ( p(a)=p(b)=p(c)=0, ) then the value of ( (2-a)(2-b)(2-c) ) is

(A) 3

(B) 5

(C) 7

(D) 9

Solution

[

begin{array}{c}

p(x)=x^{3}-3 x^{2}+2 x+5

p(a)=p(b)=p(c)=0

end{array}

]

( Rightarrow quad x=a, b, c ) are

rosts of polynomial ( p(x) )

[

begin{array}{l}

Rightarrow p(x)=(x-a)(x-b)(x-c)

text { wher, } quad a+b+c=3

quad a b+b c+a c=2

quad a b c=-5

begin{aligned}

b(2)=&(2-a)(2-b)(2-c)

=& 2^{3}-3(2)^{2}+4+5

&=8-12+4+5

&=5

end{aligned}

end{array}

]

Option B.