Check whether polynomial ( x – 1) is a factor of the polynomial (x3– 8x2+19x –12).Verify by division algorithm.
Answers
Answer:
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Step-by-step explanation:
Given :-
The polynomial x³– 8x²+19x –12
To find :-
Check whether polynomial (x-1) is a factor of the polynomial (x³- 8x²+19x -12).
Verify by division algorithm.
Solution :-
Given Cubic Polynomial is
p(x) =x³-8x²+19x -12
Given linear polynomial = (x-1)
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
=> (1)³-8(1)²+19(1)-12
=> 1-8(1)+19-12
=> 1-8+19-12
=> (1+19)+(-8-12)
=> 20+(-20)
=> 20-20
=> 0
So we have p(1) = 0
Therefore, (x-1) is a factor of p(x).
Verification:-
Division Algorithm on Polynomials is
p(x) = g(x)×q(x)+r(x)
Where, p(x) is the given Polynomial or dividend
g(x) = divisor
q(x) = quotient
r(x) = remainder
(x-1) is a factor of p(x) then the remainder is 0
x-1)x³-8x²+19x-12(x²-7x+12
x³-x²
(-)
_____________
0 -7x²+19x
-7x²+7x
(+)
______________
0 +12x-12
12x-12
_______________
0
_______________
So , the remainder is zero .
and
p(x) = g(x)×q(x)+r(x)
=>(x-1) (x²-7x+12)+0
=> x(x²-7x+12)-1(x²-7x+12)
=> x³-7x²+12x-x²+7x-12
=> x³-8x²+19x-12
Verified the given relations in the given problem.
Used formulae:-
Division Algorithm on Polynomials is
p(x) = g(x)×q(x)+r(x)
Where, p(x) is the given Polynomial or dividend
g(x) = divisor
q(x) = quotient
r(x) = remainder
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.