Question

find the zero of the polynomial f(x)=4√3x^2+5x-2√3x
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Answer 1

Answer:

Find the zeros of the polynomial f(x) =4√3x²+5x-2√3 verify the relationship between the zeros & its coefficients?

The given polynomial f(x)

= 4√3 x^2 +5x - 2√3

= 4√3 x^2 + 8x - 3x - 2√3

= 4x(√3 x +2) - √3 (√3 x +2)

= (√3x+2)(4x-√3)

Hence the zeroes are -2/√3 and √3/4

Sum of the roots = -2/√3 + √3/4 = -2√3/3 + √3/4 = (-8√3+3√3)/12 = -5√3/12

Sum of the roots = -b/a = -5/4√3 = -5√3/12

Hence sum of the roots =-b/a.

Product of the roots = (-2/√3)(√3/4) = -1/2.

Product of the roots = c/a = -2√3/4√3 = -1/2.

So product of the roots = c/a.

Thus, the relationship between the roots and coefficients is verified.

Answer 2

Answer:

solution ; let the polynomial be given by f(x)

f(x) =

$4 \sqrt{3} x ^{2} + 5x - 2 \sqrt{3} \\ \\ 4 \sqrt{3} x ^{2} + 8x - 3x - 2 \sqrt{3} \\ \\ 4x( \sqrt{3}x + 2) - \sqrt{3} ( \sqrt{3}x + 2) \\ \\ (4x - \sqrt{3} )( \sqrt{3} x+ 2) \\ \\ x =$

zeroes of polynomial is =

$x = \frac{ \sqrt{3} }{4} \: \: and \: x = \frac{2}{ \sqrt{3} }$