How do you write a polynomials of least degree with integer coefficients that has the given zeros in expanded form?

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1 Answer
Dec 4, 2017

#x^3-2x^2-8x+16#

Explanation:

For polynomials, a radical zero cannot occur in isolation; if one of the zeros is #2sqrt(2)#, then the conjugate zero #-2sqrt(2)# must also be present. Thus, we know there are 3 factors:

#(x-2)(x-2sqrt(2))(x+2sqrt(2))#

By multiplying this out we can derive the polynomial. It can be helpful to multiply out the final 2 terms so that the radical is removed:

#(x-2sqrt(2))(x+2sqrt(2)) = x^2+2xsqrt(2)-2xsqrt(2)-4*sqrt(4)#

# = x^2-8#

Finally:

#(x-2)(x^2-8) = x^3 -8x - 2x^2 + 16#

# = x^3-2x^2-8x+16#