Question

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

Answer 1

x + 1 and x -1 are the factors of

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

Let x + 1 = 0, then = - 1

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

= - a + 1 + 2 + b

= 3 - a + b

$\therefore$ x + 1 is the factor of f(x)

$\therefore$ Remainder = 0

$\therefore$ 3 - a + b = 0 $\Rightarrow$ a - b = 3

Again if x - 1 = 0, then x = 1

$\therefore f\left(x\right)=a(1{)}^3+(1{)}^2-2\times$ 1 + b

= a + 1 -2 + b = a + b -1

$\therefore$ x - 1 is the factor of f(x)

$\therefore$ Remainder = 0

$\therefore$ a + b - 1 = 0 $\Rightarrow$ a + b = 1

Adding we get 2a = 4 $\Rightarrow a=\frac{4}{2}=2$

and subtracting, 2b = - 2

$\Rightarrow b=\frac{-2}{2}=-1$

$\therefore$ a = 2, b = - 1