Question

Find the zero of the polynomial y2+4root3y-15 find the relationship between zeroes and co-offecient

Answer 1

$\sf{GIVEN:-}$ y² + 4√3y - 15 = 0
Where, cofficient of x² = 1 , cofficient of x = 4√3 , constant or cofficient of x⁰ = -15

now finding the zeroes of the polynomial by splitting middle term.

y² + 4√3y - 15 = 0

y² + 5√3y - √3y - 15 = 0

y(y + 5√3) - √3(y + 5√3) = 0

(y - √3)(y + 5√3) = 0

y = √3 , y = -5√3

Now,

$\sf{\bold{RELATIONS\:\: BETWEEN\:\: ZEROES\:\: AND\:\: CO-OFFIENT}}$

sum of zeroes = (-cofficient of x) / (cofficient of x²)

√3 + (-5√3) = -4√3/1

-4√3 = -4√3


product of zeroes = cofficient of x⁰/cofficient of x²

(√3) * (-5√3) = (-15)/1

-15 = -15

$\huge{\sf{\bold{HENCE, \:\: VERIFIED}}}$

Answer 2

$Given : f(x) = y^2 + 4 \sqrt{3}y - 15$

Zero of the polynomial is the value of x. Substitute f(x) = 0

$= \ \textgreater \ y^2 + 4 \sqrt{3}y - 15 = 0$

$= \ \textgreater \ y^2 + 5 \sqrt{3}y - \sqrt{3}y - 15 = 0$

$= \ \textgreater \ y(y + 5\sqrt{3} )- \sqrt{3} (y + 5 \sqrt{3}) = 0$

$= \ \textgreater \ (y - \sqrt{3})(y + 5 \sqrt{3} ) = 0$

$= \ \textgreater \ y = \sqrt{3}, -5 \sqrt{3}$


The verification is explained in the attachment.


Hope it helps!