find all zeroes of the polynomial 2x4+7x3-19x2-14x+30 if two if its zeroes are √2 and -√2
Answers
Step-by-step explanation:
Given:-
2x^4+7x^3-19x^2-14x+30 has two zeroes √2 and -√2.
To find:-
Find all zeroes of the polynomial ?
Solution:-
Given bi-quadratic polynomial :
P(x) = 2x^4+7x^3-19x^2-14x+30
Given zeroes are √2 and -√2
We know that
√2 is a zero of P(x) then (x-√2) is a factor.
-√2 is a zero of P(x) then (x+√2) is a factor.
=> (x-√2)(x+√2) is also a factor of P(x).
=> x^2-(√2)^2 is aslo a factor of P(x).
=> x^2-2 is aslo a factor of P(x).
To get other zeroes we have to divide P(x) by (x^2-2).
2x^2+7x-15
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x^2-2 ) 2x^4+7x^3-19x^2-14x+30
2x^4 -4x^2
(-) (+)
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0 + 7x^3 -15x^2 -14x
7x^3 -14x
(-). (+)
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0 -15x^2 +0+30
-15x^2 + 30
(+). (-)
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0 - Remainder
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Quotient = 2x^2+7x-15
Remainder = 0
P(x) = 2x^4+7x^3-19x^2-14x+30
=>(x^2-2)( 2x^2+7x-15)
To get zeores we write P(x) = 0
=> (x^2-2)( 2x^2+7x-15) = 0
=> (x+√2)(x-√2)(2x^2+10x-3x-15)=0
=> (x+√2)(x-√2)[2x(x+5)-3(x+5)] =0
=> (x+√2)(x-√2)(x+5)(2x-3) = 0
=> (x+√2)=0 or (x-√2)=0 or(x+5)=0 or 2x-3=0
=>x= -√2 or x=√2 or x=-5 or 2x = 3
=> x= -√2 or x=√2 or x=-5 or x=3/2
Zeroes are -√2 ,√2 , -5 , 3/2
Answer:-
The other zeroes of P(x) are -5 and 3/2
Used Concept :-
- To get zeroes of the given Polynomial we equate it zero.