Question

If both x + 1 and x - 1 and factors of ax3 + x2-2x + b, find the values of a and b​

Answer 1

Refer The Attachment ⬆️

Value of a and b

• 2 & -1

$\bold{@MrMonarque}$

Hope It Helps You ✌️

Answer 2

$\large\underline{\sf{Given- }}$

$\rm :\longmapsto\:(x + 1) \: and \: (x - 1) \: are \: factors \: of \: {ax}^{3} + {x}^{2} - 2x + b$

Let assume that

$\rm :\longmapsto\:f(x) \: = \: {ax}^{3} + {x}^{2} - 2x + b$

Now,

$\rm :\longmapsto\:x + 1 \: is \: a \: factor \: of \: f(x)$

We know,

Factor Theorem states that if x - a is a factor of polynomial f (x), then f(a) = 0.

So,

$\rm :\longmapsto\:f( - 1) = 0$

$\rm :\longmapsto\: \: {a( - 1)}^{3} + {( - 1)}^{2} - 2( - 1) + b = 0$

$\rm :\longmapsto\: - a + 1 + 2 + b = 0$

$\rm :\longmapsto\: - a + 3 + b = 0$

$\bf\implies \:a = 3 + b - - - - (1)$

Now,

$\rm :\longmapsto\:x - 1 \: is \: a \: factor \: of \: f(x)$

By factor theorem,

$\rm :\longmapsto\:f(1) = 0$

$\rm :\longmapsto\: \: {a(1)}^{3} + {(1)}^{2} - 2(1) + b = 0$

$\rm :\longmapsto\:a + 1 - 2 + b = 0$

$\rm :\longmapsto\:a - 1 + b = 0$

$\rm :\longmapsto\:b + 3 - 1 + b = 0 \: \: \: \: \: \{ \: using \: (1) \}$

$\rm :\longmapsto\:2b + 2 = 0$

$\rm :\longmapsto\:2b= - 2$

$\bf :\longmapsto\:b \: = - \: 1$

On substituting b = - 1 in equation (1), we get

$\rm :\longmapsto\:a = 3 - 1$

$\bf :\longmapsto\:a = 2$

$\begin{gathered}\begin{gathered}\bf\: \ :\longmapsto\:Hence-\begin{cases} &\bf{a \: = \: 2} \\ &\bf{b \: = \: - \: 1} \end{cases}\end{gathered}\end{gathered}$

Additional Information :-

Remainder Theorem :-

This theorem states that if a polynomial f(x) is divided by linear polynomial (x - a), then remainder is f(a).