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Question :-
If x³ + ax² - bx + 10 is divisible by x² - 3x + 2 , find the values of a + b
Answer :-
Given :-
x³ + ax² - bx + 10 is divisible by x² - 3x + 2
Required to find :-
- Values of a + b ?
Solution :-
Given information :-
x³ + ax² - bx + 10 is divisible by x² - 3x + 2
we need to find the value of " a + b "
So
Let's consider ,
p ( x ) = x³ + ax² - bx + 10
&
g ( x ) = x² - 3x + 2
Since, g ( x ) is a quadratic expression
Let's Factorise it !
x² - 3x + 2
x² - 2x - 1x + 2
x ( x - 2 ) - 1 ( x - 2 )
( x - 1 ) ( x - 2 )
Hence,
( x - 1 ) ( x - 2 ) can divide p ( x )
So,
Let,
x - 1 = 0
x = 1
Substitute this value in place of x in p ( x )
So,
p ( 1 ) =
( 1 )³ + a ( 1 )² - b ( 1 ) + 10 = 0
1 + a - b + 10 = 0
11 + a - b = 0
- b = - a - 11
Taking minus ( - ) common on both sides
- ( b ) = - ( a + 11 )
=> b = a + 11
Consider this as equation 1
Similarly,
Let,
x - 2 = 0
x = 2
p ( 2 ) =
( 2 )³ + a ( 2 )² - b ( 2 ) + 10 = 0
8 + a ( 4 ) - 2b + 10 = 0
8 + 4a - 2b + 10 = 0
18 + 4a - 2b = 0
Substitute the value of b from Equation 1
18 + 4a - 2 ( a + 11 ) = 0
18 + 4a - 2a - 22 = 0
2a - 4 = 0
2a = 4
a = 4/2
=> a = 2
Substitute the value of a in equation 1
So,
b = a + 11
b = 2 + 11