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TASSpecialist MathematicsQuick questions

Unit 3

Quick questions on Factorising polynomials over the complex numbers - TCE Mathematics Specialised (Tasmania)

3short Q&A pairs drawn directly from our worked dot-point answer. For full context and worked exam questions, read the parent dot-point page.

What is the conjugate root theorem?
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The reason is the conjugate arithmetic from earlier work: taking the conjugate of the whole equation P(z)=0P(z) = 0 replaces each zz by zˉ\bar z but leaves the real coefficients unchanged, so P(zˉ)=0P(\bar z) = 0 as well. A practical consequence is that real polynomials of odd degree must have at least one real root, because complex roots are used up in pairs.
What are real quadratic factors?
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A conjugate pair of roots a±bia \pm bi corresponds to the real quadratic factor
What is factorising a given polynomial?
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When you are handed a polynomial and one complex root, divide out the corresponding real quadratic, then factor the remaining quotient.

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