If both (x + 1) and (x - 1) are the factors of ax3

Question:

If both $(x+1)$ and $(x-1)$ are the factors of $a x^{3}+x^{2}-2 x+b$, Find the values of $a$ and $b$

Solution:

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

(x + 1) and (x - 1) are the factors

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

Let, x - 1= 0

⟹ x = -1

Substitute x value in f(x)

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

= a + 1 - 2 + b

= a + b - 1 ..... 1

Let, x + 1 = 0

⟹ x = -1

Substitute x value in f(x)

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

= -a + 1 + 2 + b

= -a + b + 3 ..... 2

Solve equations 1 and 2

a + b = 1

-a + b = -3

2b = -2

⟹ b = -1

substitute b value in eq 1

⟹ a - 1 = 1

⟹ a = 1 + 1

⟹ a = 2

The values are a= 2 and b = -1

Leave a comment

Close

Click here to get exam-ready with eSaral

For making your preparation journey smoother of JEE, NEET and Class 8 to 10, grab our app now.

Download Now