To determine the number of rational roots of the function \( f(x) = x^6 - 2x^4 - 5x^2 + 6 \), we need to perform several steps:
Step 1: Identify Roots of the Polynomial
First, we need to find the roots of the polynomial \( f(x) \). The roots are the values of \( x \) for which \( f(x) = 0 \).
The roots of the polynomial \( f(x) = x^6 - 2x^4 - 5x^2 + 6 \) are:
[tex]\[ x = -1, 1, -\sqrt{3}, \sqrt{3}, -\sqrt{2}i, \sqrt{2}i \][/tex]
Step 2: Determine Rationality of Each Root
Next, we identify which of these roots are rational numbers. A rational number is any number that can be expressed as the quotient of two integers (i.e., \( \frac{a}{b} \), where \( a \) and \( b \) are integers and \( b \neq 0 \)).
- \( -1 \) is a rational number because it can be written as \( \frac{-1}{1} \).
- \( 1 \) is a rational number because it can be written as \( \frac{1}{1} \).
- \( -\sqrt{3} \) is an irrational number because the square root of a non-square integer is irrational.
- \( \sqrt{3} \) is an irrational number for the same reason as \( -\sqrt{3} \).
- \( -\sqrt{2}i \) involves the imaginary unit \( i \), making it a complex number, not a rational number.
- \( \sqrt{2}i \) also involves the imaginary unit \( i \), making it a complex number, not a rational number.
Step 3: Count the Number of Rational Roots
We count the roots that are rational:
- There are 2 rational roots: \( -1 \) and \( 1 \).
Conclusion
The number of rational roots of the polynomial [tex]\( f(x) = x^6 - 2x^4 - 5x^2 + 6 \)[/tex] is thus [tex]\( \boxed{2} \)[/tex].
