What is the polynomial function #f# of least degree that has rational coefficients, a leading coefficient of 2, and the zeros #-5, sqrt3#?

1 Answer
George C.
Nov 25, 2017

#f(x) = 2x^3+10x^2-6x-30#

Explanation:

Since we want rational coefficients, any irrational zeros must occur in radical conjugate pairs.

So the zeros are #-5#, #sqrt(3)# and #-sqrt(3)#, with corresponding factors #(x+5)#, #(x-sqrt(3))# and #(x+sqrt(3))#.

To get leading coefficient #2#, include #2# as a multiplier to find:

#f(x) = 2(x+5)(x-sqrt(3))(x+sqrt(3))#

#color(white)(f(x)) = 2(x+5)(x^2-3)#

#color(white)(f(x)) = 2(x^3+5x^2-3x-15)#

#color(white)(f(x)) = 2x^3+10x^2-6x-30#

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