The graph of f(x) = x6 – 2x4 – 5x2 + 6 is shown below. On a coordinate plane, a graph of a function is in all four quadrants. It has 2 minimum curves at (1. 5, negative 4) and (1. 5, negative 4). It has a maximum curve at (0,6) and crosses the x-axis 4 times. How many roots of f(x) are rational numbers? 1 2 4 6.

There are two rational roots for f(x)

Explanation:

We are given a function

f(x) = x^6-2x^4-5x^2+6f(x)=x6−2x4−5x2+6

To find the number of rational roots for f(x).

Let us use remainder theorem that when

f(a) =0, (x-a) is a factor of f(x) or x=a is one solution.

Substitute 1 for x

f(1) = 1-2-5+6=0

Hence x=1 is one solution.

Let us try x=-1

f(-1) = 1-2-5+6 =0

So x =-1 is also a solution and x+1 is a factor

We can write f(x) by trial and error as

f(x) = (x-1)(x+1)(x^2-3)f(x)=(x−1)(x+1)(x2−3)

We find that f(x) (x^2-3)f(x)(x2−3) factor gives two irrational solutions as

±√3.

Hence number of rational roots are 2.

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