Question

Find the value for k for which x x ^ 4 + 10 x cube + 25 x square + 15 x + k is exactly divisible by x + 7

Answer 1

p(x) = x4-x3+7x2+ax+b

q(x) = x2-1

=(x+1)(x-1)

x+1 and x-1 are factors of p(x)

now,

p(-1) =0 and p(1)=0

p(-1) = -14-(-1)3+7(-1)2+a(-1)+b =0 and p(1) = 14-13+7.12+a.1+b = 0

=>1+1+7-a+b=0 and 1-1+7+a+b=0

=>9-a+b=0 and 7+a+b=0

=>-a+b=-9.... (1) and a+b=-7.....(2)

on subtracting (1) from (2), we get == >

a+b-(-a+b)=-7-(-9)

a+b+a-b=-7+9

=>2a=2

=>a=1

putting a=1, in (1), we get,

-1+b=-9

b=9+1= 10

a=1, b=10...ans