Question

find the polynomial whose zero are square of the zero of the polynomial 3x2 + 6x -9

Answer 1

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p(x) = 3x^2+6x-9=0
p(x) = 3x^2+9x-3x-9=0
p(x) = 3x(x+3)-3(x+3)=0
p(x) = (x+3)(3x-3)=0

If x+3=0 then x=(-3)
If 3x-3=0 then 3x=3 so , x=1

So ,
$\alpha = - 3$
And

So ,
$\beta = 1$
For another polynomial ,

Sum of zeroes = alpha^2 + beta^2
So,
Sum of zeroes = (alpha + beta) ^2 - 2alpha.beta

Sum of zeroes = (-3+1)^2-2(-3)(1)

Sum of zeroes = (-2)^2+6

Sum of zeroes = 4+6=10

And ,

Product of zeroes = alpha ^2 × beta^2
So ,
Product of zeroes = (-3)^2 × (1)^2

Product of zeroes = 9×1=9

New polynomial ,

p(x) = x^2-(sum of zeroes)x + product of zeroes

p(x) = x^2 - 10x + 9


Therefore , New polynomial = x^2-10x+9 .