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Find the zeroes of the quadratic polynomial f(x) = 5x2 ˗ 4 ˗ 8x and verify the relationship between the zeroes and coefficients of the given polynomial.

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We have: 

f(x) = 5x2 ˗ 4 ˗ 8x 

= 5x2 ˗ 8x ˗ 4 

= 5x2 ˗ (10x ˗ 2x) ˗ 4 

= 5x 2 ˗ 10x + 2x ˗ 4 

= 5x (x ˗ 2) + 2(x ˗ 2) 

= (5x + 2) (x ˗ 2) 

∴ f(x) = 0 ⇒ (5x + 2) (x ˗ 2) = 0 

⇒ 5x + 2= 0 or x ˗ 2 = 0

⇒ x = \(\frac{-2}5\) or x = 2

So, the zeroes of f(x) are \(\frac{-2}5\) and 2.

Sum of zeroes = \((\frac{-2}5)\) + 2 = \(\frac{-2+10}5\) = \(\frac{8}5\) = \(\frac{-(coefficient\,of\,x)}{(coefficient\,ofx^2)}\)

Product of zeroes = \((\frac{-2}5)\) x 2 = \(\frac{-4}5\) = \( \frac{constant\,term}{(coefficient\,of\,x^2)}\)

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