Q14 of 30 Page 1

Find all zeroes of the polynomial (2x4 – 9x3 + 5x2 + 3x – 1) if two of its zeroes are (2 + √3) and (2 – √3)

Given:

• f(x) = (2x4 – 9x3 + 5x2 + 3x – 1)


• Zeroes = (2 + √3) and (2 – √3)


Given the zeroes, we can write the factors = (x – 2 + √3) and (x – 2 – √3)


{Since, If x = a is zero of a polynomial f(x), we can say that x - a is a factor of f(x)}


Multiplying these two factors, we can get another factor which is:


((x – 2) + √3)((x – 2) – √3) = (x – 2)2 – (√3)2


(Using (a2 – b2 ) = (a + b)(a – b))


⇒x2 + 4 – 4x – 3 = x2 – 4x + 1


So, dividing f(x) with (x2 – 4x + 1)


f(x) = (x2 – 4x + 1) (2x2 – x – 1)


Solving (2x2 – x – 1), we get the two remaining roots as:



where f(x) = ax2 + bx + c = 0


(using Quadratic Formula)





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