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Answer 1

Question : -

The polynomials x³ + 2x² - 5ax - 8 and x³ - ax² - 12x - 6 when divided by ( x - 2 ) & ( x - 3 ) respectively leaves remainder p and q . If q - p = 10 , Find the value of a ?

Answer : -

Given : -

Polynomials x³ + 2x² - 5ax - 8 and x³ - ax² - 12x - 6 when divided by ( x - 2 ) & ( x - 3 ) respectively leaves remainder p and q . If q - p = 10 .

Required to find : -

• Value of a ?

Solution : -

Polynomials x³ + 2x² - 5ax - 8 and x³ - ax² - 12x - 6 when divided by ( x - 2 ) & ( x - 3 ) respectively leaves remainder p and q . If q - p = 10 .

The given polynomials are Cubic Polynomials;

Cubic Polynomials ,

x³ + 2x² - 5ax - 8 and x³ - ax² - 12x - 6

Let's consider this as ,

p ( x ) = x³ + 2x² - 5ax - 8

q ( x ) = x³ - ax² - 12x - 6

It is mentioned that when p ( x ) is divided by ( x - 2 ) it leaves remainder p . Similarly, when q ( x ) is divided by ( x - 3 ) it leaves remainder q .

So,

Let ,

x - 2 = 0

x = 2

p ( 2 ) =

( 2 )³ + 2 ( 2 )² - 5a ( 2 ) - 8 = p

8 + 8 - 10a - 8 = p

- 8 , + 8 get's cancelled due to the opposite signs

8 - 10a = p

This implies ;

p = 8 - 10a $\longrightarrow{\tt{\red{Equation - 1 }}}$

consider this as equation 1

Similarly,

Let,

x - 3 = 0

x = 3

g ( 3 ) =

( 3 )³ - a ( 3 )² - 12 ( 3 ) - 6 = q

27 - 9a - 36 - 6 = q

27 - 9a - 42 = q

- 15 - 9a = q

q = - 15 - 9a $\longrightarrow{\tt{\red{Equation - 2 }}}$

Consider this as equation - 2

Now,

It is also mentioned that ;

q - p = 10

Here,

Substitute the respective values of p , q from equations 1 & 2

This implies ;

- 15 - 9a - ( 8 - 10a ) = 10

- 15 - 9a - 8 + 10a = 10

- 23 + a = 10

a = 10 + 23

a = 33

Therefore,

Value of a = 33

Additional Information : -

What is remainder theorem ?

Remainder theorem can be simply stated as ;

If p ( x ) when divided by ( x - a ) it leaves remainder . The remainder which is left is exactly equal to the value of p ( a ) .

Example : -

x² - 3x + 2 when divided by ( x + 1 ) leaves remainder

So,

x + 1 ) x² - 3x + 2 ( x - 4

.......... x² + x

..........(-)...(-)....

..................- 4x + 2

..................- 4x - 4

..................(+)...(+).....

..........................6 .....

The remainder is 6

p ( - 1 ) =

( - 1 )² - 3 ( - 1 ) + 2

1 + 3 + 2

6

value of p ( - 1 ) = 6

Remainder theorem allows us to find the remainder instead of performing the long division .