Which is a third degree polynomial with #-1# and #1# as its only zeros? a. #x^3 - x^2 - x + 1# b. #x^3 - x^2 + x - 1# c. #x^3 - 3x^2 + 3x - 1# d. #x^3 + 3x^2 + 3x + 1#

1 Answer
George C.
Nov 10, 2017

a. #x^3-x^2-x+1#

Explanation:

a. #bb(x^3-x^2-x+1)#

#x^3-x^2-x+1 = (x^3-x^2)-(x-1)#

#color(white)(x^3-x^2-x+1) = x^2(x-1)-1(x-1)#

#color(white)(x^3-x^2-x+1) = (x^2-1)(x-1)#

#color(white)(x^3-x^2-x+1) = (x-1)(x+1)(x-1)#

Zeros: #1# (with multiplicity #2#) and #-1#

b. #bb(x^3-x^2+x-1)#

#x^3-x^2+x-1 = (x^3-x^2)+(x-1)#

#color(white)(x^3-x^2+x-1) = x^2(x-1)+1(x-1)#

#color(white)(x^3-x^2+x-1) = (x^2+1)(x-1)#

#color(white)(x^3-x^2+x-1) = (x-i)(x+i)(x-1)#

Zeros: #+-i# and #1#

c. #bb(x^3-3x^2+3x-1)#

#x^3-3x^2+3x-1 = (x-1)^3#

Zeros: #1# with multiplicity #3#

d. #bb(x^3+3x^2+3x+1)#

#x^3+3x^2+3x+1 = (x+1)^3#

Zeros: #-1# with multiplicity #3#

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