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