Question

Find a quadratic poly, the sum of zeroes is 0 and one zero is 5

Answer 1

let,
the roots of the required polynomial be   α and β
given, α = 5
sum of the roots = 0
         ⇒ α + β = 0
         ⇒ 5 + β  = 0
         ⇒ β = -5
therefore, the roots of the polynomial are 5 and -5
the quadratic polynomial with roots α and β is  x²- (α+β)x + αβ= 0
here, α =5 ,β = -5
required quadratic polynomial = x² -(5-5)x + 5(-5)
                                                 = x² -0 -25
                                                 = x²-25

Answer 2

$\alpha + \beta = 0 \\ \alpha = 5 \\ \beta = - 5 \\ \alpha \beta = - 25 \\ required \: polynomial \\ = k( {x}^{2} - x( \alpha + \beta ) + \alpha \beta ) \\ = {x}^{2} - 0x - 25 \\ = {x}^{2} - 25$