sum of 2 zeroes of a polynomial of degree 4 is -1 and their product is -2.If other two zeroes are root 3 and -root 3.find the polynomial.  

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lfryerda | High School Teacher | (Level 2) Educator

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Two of the zeros are `sqrt 3` and `-sqrt3` which means that two factors of the polynomial are `x-sqrt3` and `x+sqrt3` .  The remaining two factors must then be `x-r_1` and `x-r_2` where `r_1` and `r_2` are the other roots.  By expanding these factors, we get:

`(x-r_1)(x-r_2)`

`=x^2-(r_1+r_2)x+r_1r_2`   

But we are told that the sum of the other two zeros are -1 and their product is -2, which means that this quadratic becomes:

`x^2+x-2`

Multiplying by the first two factors, we get the polynomial up to an arbitrary scaling factor to get

`f(x)=a(x-sqrt3)(x+sqrt3)(x^2+x-2)`

`=a(x^2-3)(x^2+x-2)`

`=a(x^4+x^3-5x^2-3x+6)`

where `a ne 0` is any real number.

The polynomial is `f(x)=a(x^4+x^3-5x^2-3x+6)` .

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