Find the values of a and b so that (x + 1) and (x – 1) are the factors of
Question:

Find the values of $a$ and $b$ so that $(x+1)$ and $(x-1)$ are the factors of $x^{4}+a x^{3}-3 x^{2}+2 x+b$

Solution:

Here, $f(x)=x^{4}+a x^{3}-3 x^{2}+2 x+b$

The factors are (x + 1) and (x – 1)

From factor theorem, if x = 1, -1 are the factors of f(x) then f(1) = 0 and f(-1) = 0

Let, us take x + 1

⟹ x + 1 = 0

⟹ x = -1

Substitute value of x in f(x)

$f(-1)=(-1)^{4}+a(-1)^{3}-3(-1)^{2}+2(-1)+b$

= 1 – a – 3 – 2 + b

= -a + b – 4 … 1

Let, us take x – 1

⟹ x – 1 = 0

⟹ x = 1

Substitute value of x in f(x)

$f(1)=(1)^{4}+a(1)^{3}-3(1)^{2}+2(1)+b$

= 1 + a – 3 + 2 + b

= a + b …. 2

Solve equations 1 and 2

– a + b = 4

a + b = 0

2b = 4

b = 2

substitute value of b in eq 2

a + 2 = 0

a = – 2

the values are a = -2 and b = 2