Obtain all the zeroes of 3x⁴+6x³-2x²-10x-5
Answers
Hi there!
Please refer to the attachments for answer..
Given p(x) = 3x⁴+6x³-2x²-10x-5--(1)
The degree of p(x) is 4 , so it has at most 4 zeroes.
The two zeroes of given p(x) are
√(5/3) and -√(5/3)
The equation of the polynomial whose zeroes are √(5/3) and -√(5/3 )
= ( x+√5/3 )( x - √5/3 )
= x² - 5/3 --- (2)
Divide (1) with ( 2 ) , we get
x²-5/3)3x⁴+6x³-2x²-10x-5(3x²+6x+3
3x⁴+ 0 - 5x²
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6x³+3x²-10x-5
6x³ + 0 - 10x
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3x² + 0 - 5
3x² + 0 - 5
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( 0 )
p(x) = ( x²-5/3)( 3x²+6x+3 )
= 3(x²-5/3)( x²+2x+1 )
= 3[ x² - {√(5/3)}² ] ( x + 1 )²
= 3(x+√5/3)(x-√5/3)(x+1)(x+1)
Therefore ,
The zeroes of the given polynomial
are -√(5/3), √(5/3) , -1, -1