Question

Find the value of a and b so that the polynomial x^3-4x^2+ax+b is exactly divisible bh x-2 as well as x+1

Answer 1

Answer:

a = 1

b = 6

Note:

★ If (x-a) is a factor of any polynomial p(x) , then x = a is a zero of p(x) and p(a) = 0 .

Solution:

Let the given polynomial be p(x) .

Thus,

p(x) = x³ - 4x² + ax + b

Also,

It is given that , x - 2 is a factor of the given polynomial p(x) .

Thus,

=> p(2) = 0

=> 2³ - 4•2² + a•2 + b = 0

=> 8 - 16 + 2a + b = 0

=> - 8 + 2a + b = 0

=> 2a + b = 8 ----------(1)

Also,

It is given that , x + 1 is a factor of the given polynomial p(x) .

Thus,

=> p(-1) = 0

=> (-1)³ - 4(-1)² + a(-1) + b = 0

=> - 1 - 4 - a + b = 0

=> - 5 - a + b = 0

=> - a + b = 5 ---------(2)

Now,

Subtracting eq-(2) from eq-(1) , we get ;

=> 2a + b - (-a + b) = 8 - 5

=> 2a + b + a - b = 3

=> 3a = 3

=> a = 3/3

=> a = 1

Now,

Putting a = 1 in eq-(2) , we get ;

=> - a + b = 5

=> -1 + b = 5

=> b = 5 + 1

=> b = 6

Hence,

a = 1 and b = 6