if the polynomial (x^6+2x^3+8x^2+12x+18) is divided by another polynomial (x^2+5) , the remainder comes out to be (px+q). find the values of p and q
Answers
Answer:
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Step-by-step explanation:
Given :-
The polynomial (x⁶+2x³+8x²+12x+18) is divided by another polynomial (x²+5) , the remainder comes out to be (px+q).
To find:-
Find the values of p and q ?
Solution :-
Given Polynomial = P(x) = x⁶+2x³+8x²+12x+18
Given divisor = g(x) = x²+5
On dividing P(x) by g(x) then
x⁴-5x²+2x+33
___________________
x²+5 ) x⁶+0x⁴+2x³+8x²+12x+18
x⁶+5x⁴
(-) (-)
___________________
-5x⁴+2x³+8x²
-5x⁴ -25x²
(+) (+)
___________________
2x³ +33x²+12x
2x³ +10x
(-) (-)
___________________
33x² +2x+18
33x² + 165
(-) (-)
_____________________
2x -147
______________________
Quotient = q(x) = x⁴-5x²+2x+33
Remainder = 2x-147
According to the given problem
Remainder = px+q
=> 2x-147 = px+q
=> 2x+(-147) = px+q
On Comparing both sides then
=> p = 2 and q = -147
Answer:-
The values of p and q are 2 and -147 respectively.
Check:-
Fundamental Theorem on Polynomials is
p(x) = g(x)×q(x) + r(x)
=> (x²+5)×(x⁴-5x²+2x+33) +(2x-147)
=> x²(x⁴-5x²+2x+33)+5(x⁴-5x²+2x+33)+2x-147
=> x⁶-5x⁴+2x³+33x²+5x⁴-25x²+10x+165+2x-147
=> x⁶+2x³+8x²+12x+18
=> p(x)
Verified the given relations in the given problem
Used Method :-
- Long Division method
Used formulae:-
Fundamental Theorem on Polynomials is
p(x) = g(x)×q(x) + r(x).
Where,
- p(x) = Dividend
- g(x) = Divisor
- q(x) = Quotient
- r(x) = Remainder