Math, asked by kashisharora9999, 10 months ago

Send me question no. 10

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Answers

Answered by MisterIncredible
5

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 \tt{\longrightarrow{Equation -1 }}

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

=> b = 13

Therefore,

Value of ' a + b ' = 2 + 13 = 15

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