find the values of a and b ,if (x-1) and (x-2) are the factors of x^3-ax^2+bx-8
Answers
Step-by-step explanation:
Given :-
(x-1) and (x-2) are the factors of x³-ax²+bx-8
To find :-
Find the values of a and b ?
Solution :-
Given Cubic Polynomial is x³-ax²+bx-8
Let P(x) = x³-ax²+bx-8
We know that
Factor Theorem: If x-a is a factor of P(x) then P(a) = 0
If (x-1) is a factor of P(x) then P(1) = 0
=> P(1) = (1)³-a(1)²+b(1)-8 = 0
=> 1-a(1)+b-8 = 0
=> 1-a+b-8 = 0
=>-a+b-7 = 0
=> b = a+7 ------------------------(1)
And
If (x-2) is a factor of P (x) then P(2) = 0
=> (2)³-a(2)²+b(2)-8 = 0
=> 8-a(4)+2b-8 = 0
=> 8-4a+2b-8 = 0
=> -4a+2b = 0
=> -2(2a-b) = 0
=> 2a-b = 0/-2
=> 2a-b = 0
=> 2a = b -------------------------(2)
From (1)&(2)
2a = a+7
=> 2a-a = 7
=> a = 7
Therefore, a = 7
On Substituting the value of a in (2) then
=> b = 2(7)
=> b = 14
Therefore, b = 14
Answer:-
The values of a and b are 7 and 14 respectively.
Check:-
If a = 7 and b = 14 then the Cubic Polynomial becomes x³-7x²+14x-8
=> x³-6x²-x²+6x+8x-8
=> (x³-x²)-(6x²-6x)+(8x-8)
=> x²(x-1)-6x(x-1)+8(x-1)
=> (x-1)(x²-6x+8)
=> (x-1)(x²-2x-4x+8)
=> (x-1)[x(x-2)-4(x-2)]
=> (x-1)(x-2)(x-4)
So, x-1 and x-2 are the factors of the given Polynomial.
Verified the given relations in the given problem.
Used formulae:-
Factor Theorem:-
Let P(x) be a polynomial of the degree greater than or equal to 1 and x-a is another linear polynomial if x-a is a factor of P (x) then P(a) = 0 vice-versa.
Answer:
Step-by-step explanation:
(x-1) and (x-2)
Are the factors
Hope it helps yaa