What are all the possible rational zeros for #f(x)=x^4-7x^2+10# and how do you find all zeros?

1 Answer
Nov 20, 2016

Answer:

#f(x)# only has irrational zeros: #+-sqrt(5)# and #+-sqrt(2)#

Explanation:

Given:

#f(x) = x^4-7x^2+10#

By the rational root theorem, any rational zeros of #f(x)# are expressible in the form #p/q# for integers #p, q# with #p# a divisor of the constant term #10# and #q# a divisor of the coefficient #1# of the leading term.

That means that the only possible rational zeros are:

#+-1, +-2, +-5, +-10#

However, note that we can factor #f(x)# as a quadratic in #x^2# then using the difference of squares identity to find:

#x^4-7x^2+10 = (x^2-5)(x^2-2)#

#color(white)(x^4-7x^2+10) = (x^2-(sqrt(5))^2)(x^2-(sqrt(2))^2)#

#color(white)(x^4-7x^2+10) = (x-sqrt(5))(x+sqrt(5))(x-sqrt(2))(x+sqrt(2))#

So the only zeros of #f(x)# are irrational: #+-sqrt(5)#, #+-sqrt(2)#