Exercise 50 Page 350

3i, - 7i

Practice makes perfect

We are told that the given roots are roots of a polynomial function P(x) that has rational coefficients. - 3i and 7iTo find an additional root of P(x)=0, let's recall the second part of the 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 a complex root.

Hypotheses	Conclusion
P(x) has rational, real coefficients	3i is also a root of P(x)
- 3i is a complex root of P(x)	3i is also a root of P(x)
P(x) has rational, real coefficients	- 7i is also a root of P(x)
7i is a complex root of P(x)	- 7i is also a root of P(x)

The two additional roots that we can know for certain using the Conjugate Root Theorem are 3i and - 7i.

Exercises p.347-352

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