Question

Find the value of a and b if $x^3+10x^2+ax+b$ is divisible by both x-1 and x-2.

Answer 1

Step-by-step explanation:

Given :-

x^3+10x^2+ax+b is divisible by both x-1 and x-2

To find:-

Find the value of a and b?

Solution:-

Given cubic polynomial is x^3+10x^2+ax+b

Let P(x) = x^3+10x^2+ax+b

If P(x) is divisible by (x-1) then is satisfies the given polynomial.i.e.p(1) = 0

=> (1)^3+10(1)^2+a(1)+b=0

=> 1+10(1)+a+b = 0

=> 1+10+a+b = 0

=>11+a+b = 0

a = -(b+11)------------(1)

and

If P(x) is divisible by (x-2) then is satisfies the given polynomial.i.e.p(2) = 0

=> (2)^3+10(2)^2+a(2)+b = 0

=> 8+10(4)+2a+b = 0

=> 8+40+2a+b = 0

=> 48 +2a+b = 0

=> 48 +2[-(b+11)]+b = 0

=> 48-22-2b+b = 0

=>26 -b =0

=> b = 26

On substituting the value of b in (1) then

a = -(b+11)

=> a = -(26+11)

=> a = -37

Therefore, a= -37 and b = 26

Answer:-

The values of a and b are -37 and 26 respectively.

Used formulae:-

• If P(x) is divisible by x-a then p(a)=0 it is called Factor Theorem.