Question

given that (x+1) and(x-2) are factors o f x^3+ax^2-bx-6, find the values of a and b, factorise the given expression completely.​

Answer 1

Given :

• Two factors of the expression i.e, x³ + ax² - bx - 6 = (x + 1) and (x - 2).

To find :

The value of a and the value of b.

Solution :

From the factors of the expression , we get that :

$:\implies \bf{x + 1 = 0}$

$:\implies \bf{x = (-1)}$

$\boxed{\therefore \bf{x = (-1)}}$

$:\implies \bf{x - 2 = 0}$

$:\implies \bf{x = 2}$

$\boxed{\therefore \bf{x = 2}}$

Hence the value of x is 2 and -1.

By substituting the value of x in the equation we get :

When the value of x is 2 :

$:\implies \bf{x^{3} + ax^{2} - bx + 6 = 0}$

$:\implies \bf{2^{3} + a \times 2^{2} - b \times 2 + 6 = 0}$

$:\implies \bf{2^{3} + a \times 4 - b \times 2 + 6 = 0}$

$:\implies \bf{8 + 4a - 2b + 6= 0}$

$:\implies \bf{14 + 4a - 2b = 0}$

$:\implies \bf{4a - 2b = -14}$

$:\implies \bf{2(2a - b) = -14}$

$:\implies \bf{2a - b = \dfrac{-14}{2}}$

$:\implies \bf{2a - b = -7}$

$\boxed{\therefore \bf{2a - b = -7}}$⠀⠀⠀⠀⠀⠀Eq.(i)

When the value of x is (-1) :

$:\implies \bf{x^{3} + ax^{2} - bx + 6 = 0}$

$:\implies \bf{(-1)^{3} + a \times (-1)^{2} - b \times (-1) + 6 = 0}$

$:\implies \bf{(-1)^{3} + a \times 1 - b \times (-1) + 6 = 0}$

$:\implies \bf{-1 + a + b + 6 = 0}$

$:\implies \bf{a + b + 5 = 0}$

$:\implies \bf{a + b = -5}$

$\boxed{\therefore \bf{a + b = -5}}$⠀⠀⠀⠀⠀⠀Eq.(ii)

By adding Eq.(ii) from Eq.(i) ,we get :

$:\implies \bf{(2a - b) + (a + b) = -7 + (-5)}$

$:\implies \bf{2a - b + a + b = -7 - 5}$

$:\implies \bf{2a - \not{b} + a + \not{b} = -7 - 5}$

$:\implies \bf{2a + a = -7 - 5}$

$:\implies \bf{3a = -12}$

$:\implies \bf{a = \not{-12}{3}}$

$:\implies \bf{a = -4}$

$\boxed{\therefore \bf{a = -4}}$

Hence the value of a is -4.

By substituting the value of a in the Eq.(i) , we get :

$:\implies \bf{2a - b = -7}$

$:\implies \bf{2 \times (-4) - b = -7}$

$:\implies \bf{-8 - b = -7}$

$:\implies \bf{-(8 + b) = -7}$

$:\implies \bf{\not{-}(8 + b) = \not{-}7}$

$:\implies \bf{8 + b = 7}$

$:\implies \bf{b = 7 - 8}$

$:\implies \bf{b = -1}$

$\boxed{\therefore \bf{b = -1}}$

Hence the value of b is -1.