Question

Find the value of a and b so that x+1 and x-1 is a factor of x⁴+ax³+2x²-3x+b

Answer 1

Answer:

a = 3

b = -3

Note:

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

Solution:

Let the given polynomial be p(x) .

Thus,

p(x) = x⁴ + ax³ + 2x² - 3x + b

Also,

It is given that , (x + 1) and (x - 1) are the factors of p(x) . Thus, x = - 1 and x = 1 are the zeros of p(x) and hence p(-1) = 0 & p(1) = 0 .

Now,

=> p(-1) = 0

=> (-1)⁴ + a(-1)³ + 2(-1)² - 3(-1) + b = 0

=> 1 - a + 2 + 3 + b = 0

=> - a + b + 6 = 0

=> a = b + 6 --------(1)

Also,

=> p(1) = 0

=> (1)⁴ + a(1)³ + 2(1)² - 3(1) + b = 0

=> 1 + a + 2 - 3 + b = 0

=> a + b = 0

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

From eq-(1) and (2) , we get ;

=> b + 6 = - b

=> b + 6 + b = 0

=> 2b + 6 = 0

=> 2b = - 6

=> b = -6/2

=> b = -3

Now,

Putting b = -3 in eq-(2) , we get ;

=> a = -b

=> a = -(-3)

=> a = 3

Hence,

a = 3

b = -3