Question

If x^2-3X+2 IS a factor of x^3+ax^3+b find a and b

Answer 1

Answer :-

Given :-

g ( x ) = x² - 3x + 2

p ( x ) = x³ + ax³ + b

Required to find :-

• Values of " a " and " b " ?

Solution :-

Given that :-

g ( x ) is the factor of p ( x )

So,

Let's factorise g ( x )

g ( x ) = x² - 3x + 2

Hence,

x² - 3x + 2

x² - 2x - 1x + 2

Group the terms

x ( x - 2 ) - 1 ( x - 2 )

( x - 2 ) ( x - 1 )

Hence,

g ( x ) is factorised into ( x - 2 ) and ( x - 1 )

So,

( x - 2 ) & ( x - 1 ) are the factors of p ( x )

So,

Let ( x - 1 ) is the factor of p ( x )

x - 1 = 0

x = 1

Substitute this value in place of x in p ( x )

p ( x ) = x³ + ax³ + b

p ( 1 ) =

( 1 )³ + a ( 1 )³ + b = 0

1 + a + b = 0

b = - a - 1 $\rightarrowtail{\text{Equation - 1 }}$

Consider this as equation 1

Similarly ,

Let ( x - 2 ) is the factor of p ( x )

So,

x - 2 = 0

x = 2

p ( 2 ) =

( 2 )³ + a ( 2 )³ + b = 0

8 + a ( 8 ) + b = 0

8 + 8a + b = 0

Substitute the value of b from equation 1

So,

8 + 8a + ( - a - 1 ) = 0

8 + 8a - a - 1 = 0

7a + 7 = 0

7a = - 7

a = -7/7

a = - 1

Substitute the value of a in equation 1

b = - a - 1

b = - ( - 1 ) - 1

b = +1 -1

b = 0

Therefore ,

Values of "a" and "b" is - 1 & 0