# find the value of a and b, so that the polynomial (x3 +10x2 +ax +b) is exactly by (x-1) as well as (x-2)

let the given polynomial be

if f(x) is exactly divisible by (x-1).

if f(x) is exactly divisible by (x-2).

subtracting eq(2) by eq(1):

put a = -37 in eq(1):

thus the value of a is -37 and value of b is 26.

hope this helps you.

cheers!!

• 57

ANS: Let p(x) = x3 - 10x2 +ax +b Using factor theorem we find that if p(x) is divided by x-1 the remainder, i.e. p(1) = 0

Therefore, p(1) = (1)3 - 10(1)2 + a(1) +b = 0

=> 1 - 10 +a +b =0

=> a+ b = 9=> b = 9-a .........equation (1)
In the same way if p(x) is divided by x-2 the remainder, i.e. p(2) = 0 Therefore, p(2) = (2)3 - 10(2)2 + a(2) +b = 0

=> 8 - 40 +2a +b =0=> -32+2a+b=0...........equation (2)

Substituting the value of equation(1) in equation(2) we get, -32 + 2a+9 -a = 0

=> a = 23

Substituting the value of a in equation (1) we get b= 9- 23

=> b = -14

Hence, a=23 and b=-14

• 18

ANS: Let p(x) = x 3 + 10x 2 +ax +b Using factor theorem we find that if p(x) is divided by x-1 the remainder, i.e. p(1) = 0

Therefore, p(1) = (1)3 + 10(1)2 + a(1) +b = 0

=> 1 + 10 +a +b =0

=> a+ b = 11=> b = 11-a .........equation (1)
In the same way if p(x) is divided by x-2 the remainder, i.e. p(2) = 0 Therefore, p(2) = (2)3 + 10(2)2 + a(2) +b = 0

=> 8 + 40 +2a +b =0=> 48+2a+b=0...........equation (2)

Substituting the value of equation(1) in equation(2) we get, 48 + 2a + 11 - a = 0

=> a = -59

Substituting the value of a in equation (1) we get b= 9- (-59)

=> b = 68

Hence, a=-59 and b=68

• -7

sorry again

• -10

ANS: Let p(x) = x 3 + 10x 2 +ax +b Using factor theorem we find that if p(x) is divided by x-1 the remainder, i.e. p(1) = 0

Therefore, p(1) = (1)3 + 10(1)2 + a(1) +b = 0

=> 1 + 10 +a +b =0

=> a+ b = 11=> b = -11-a .........equation (1)
In the same way if p(x) is divided by x-2 the remainder, i.e. p(2) = 0 Therefore, p(2) = (2)3 + 10(2)2 + a(2) +b = 0

=> 8 + 40 +2a +b =0=> 48+2a+b=0...........equation (2)

Substituting the value of equation(1) in equation(2) we get, 48 + 2a - 11 - a = 0

=> a = -37

Substituting the value of a in equation (1) we get b= 9- (-59)

=> b = 46

Hence, a=-37 and b=46

• 1
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