Number of Complex Roots: six Possible Number of Real Roots: zero, two, four, or six Possible Rational Roots: ± 1/4, ± 1/2, ± 3/4, ± 1, ± 3/2, ± 2, ± 3, ± 4, ± 6, ± 8, ± 12, and ± 24
Practice makes perfect
We want to state the number of complex roots, the possible number of real roots, and the possible rational roots for the given equation.
4x^6-x^5-24=0
We will start by considering the complex roots.
If P(x) is a polynomial of degree n≥ 1, then P(x)=0 has exactly n roots.
Those roots can be either real or imaginary. Both real and imaginary numbers are complex numbers. The polynomial in the given equation has a degree of 6. Therefore, the equation has exactly six complex roots.
Real Roots
To find the possible number of real roots, let's recall the Conjugate Root Theorem for complex roots.
Conjugate Root Theorem
If P(x) is a polynomial with real coefficients, then the complex roots of P(x)=0 occur in conjugate pairs.
This theorem means that if a+bi is a root for P(x)=0, then a-bi is also a root. This means that the number of complex roots whose imaginary part is not zero is always even. Therefore, since the given equation has 6 roots, the number of complex roots whose imaginary part is not zero can be 0, 2, 4, or 6. With this information, we can find the possible number of real roots.
Possible Number of Real Roots
Number of Complex Roots - Number of Roots With Non-Zero Imaginary Part = Number of Real Roots
6- 0=6
6- 2=4
6- 4=2
6- 6=0
There are either zero, two, four, or six real roots.
Rational Roots
Finally, let's find the possible rational roots.
Recall that any rational root for P(x)=0 has the form ± p q, where p is a factor of the constant term and q is a factor of the leading coefficient.
