Find all the zeroes of the polynomial x4 − 8x3 + 19x2 − 12x +

1.	Find all the zeroes of the polynomial x4 − 8x3 + 19x2 − 12x + 2,if two of its zeroes are 2 + √2 and 2 - √2.
Answer» since two zeroes of the given polynomial are 2 + √2 and 2 - √2 , the polynomial should have two factors (x-2-√2) and (x-2+√2) that means it has quadratic factor (x-2-√2) X (x-2+√2)=(x-2)^2 -2= x²-4x+2 Now x⁴ − 8x³ + 19x² − 12x + 2 = x⁴-4x³+2x²-4x³+16x²-8x +x²-4x+2 =x²( x²-4x+2) − 4x ( x²-4x+2)+ ( x²-4x+2) =( x²-4x+2)( x²-4x+1) Hence other quadratic factor should be= (x4 − 8x3 + 19x2 − 12x + 20)/ ( x^2-4x+2) =( x²-4x+1) Solve ( x²-4x+1)=0 => (x-2)²-3=0 hence other two zeroes are 2 + √3 and 2 - √3

Find all the zeroes of the polynomial x4 − 8x3 + 19x2 − 12x + 2,if two of its zeroes are 2 + √2 and 2 - √2.

Answer»

since two zeroes of the given polynomial are 2 + √2 and 2 - √2 , the polynomial should have two factors (x-2-√2) and (x-2+√2) that means it has quadratic factor

(x-2-√2) X (x-2+√2)=(x-2)^2 -2= x²-4x+2

Now x⁴ − 8x³ + 19x² − 12x + 2

= x⁴-4x³+2x²-4x³+16x²-8x +x²-4x+2

=x²( x²-4x+2) − 4x ( x²-4x+2)+ ( x²-4x+2)

=( x²-4x+2)( x²-4x+1)

Hence other quadratic factor should be= (x4 − 8x3 + 19x2 − 12x + 20)/ ( x^2-4x+2)

=( x²-4x+1)
Solve ( x²-4x+1)=0

=> (x-2)²-3=0

hence other two zeroes are 2 + √3 and 2 - √3

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