Exercise 51 Page 350

Recall the Conjugate Root Theorem and the Complex Conjugate Root Theorem.

-2-sqrt(11), - 4+6i

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We are told that the given roots are roots of a polynomial function P(x) that has rational coefficients. -2+sqrt(11) and - 4-6i 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	- 2-sqrt(11) is also a root of P(x)
-2+sqrt(11) is an irrational root of P(x)=0	- 2-sqrt(11) is also a root of P(x)

To find the remaining root, let's recall the Complex Conjugate Root Theorem.

If P(x) is a polynomial with real coefficients, then any complex roots of P(x)=0 occur in conjugate pairs.

This means that if a + bi is a complex root, then a - bi is also a root. Let's now use this to find the other complex root.

Hypotheses	Conclusion
P(x) has rational, real coefficients	-4+6i is also a root of P(x)
- 4-6i is a complex root of P(x)	-4+6i is also a root of P(x)

The two additional roots that we can know for certain using the Conjugate Root Theorem are - 2-sqrt(11) and -4+6i.

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Exercises p.347-352

Exercise 51 Page 350

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