To find two additional roots of P(x)=0, let's recall the first part of the Conjugate Root Theorem.
If P(x) is a polynomial with rational coefficients, then any irrational roots
of P(x)=0 occur in conjugate pairs.
The above statement tell us that if a + sqrt(b) is an irrational root, then a - sqrt(b) is also a root. Let's use this to find an additional irrational root.
Hypotheses
Conclusion
P(x) has rational coefficients
5-sqrt(3) is also a root of P(x)
5+sqrt(3) is an irrational root of P(x)=0
P(x) has rational coefficients
sqrt(2) is also a root of P(x)
- sqrt(2) is an irrational root of P(x)=0
The two additional roots that we can know for certain using the Conjugate Root Theorem are 5-sqrt(3) and sqrt(2).
