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Answer:
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 }}}⟶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 }}}⟶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 .
Step-by-step explanation:
the answer of this answer is 89