Factoring with Real Number Coefficients

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Presentation transcript:

Factoring with Real Number Coefficients Sec. 2.6c Homework: p.235 37-41 odd, 45-49 odd

Factors of a Polynomial with Real Coefficients Definition: A quadratic with no real zeros is irreducible over the reals. The New Rule: Every polynomial function with real coefficients can be written as a product of linear factors and irreducible quadratic factors, each with real coefficients.

A Practice Problem Write the given function as a product of linear and irreducible quadratic factors, each with real coefficients. Graph to find a rational zero.  x = 2/3 2/3 3 –2 6 –4 –24 16 2 4 –16 3 6 –24

A Practice Problem Write the given function as a product of linear and irreducible quadratic factors, each with real coefficients. How do we factor this last term?

The zeros of the last term are complex… A Practice Problem Write the given function as a product of linear and irreducible quadratic factors, each with real coefficients. The zeros of the last term are complex… We have taken the factorization of f as far as possible, subject to the condition that each factor has real coefficients.

Polynomial Function of Odd Degree A function’s complex zeros always occur in conjugate pairs… Since a function with odd degree has an odd number of zeros, what can we conclude?... Every polynomial function of odd degree with real coefficients has at least one real zero. Does this hold with our previous example???

More Guided Practice Write the given function as a product of linear and irreducible quadratic factors, each with real coefficients.

Whiteboard Practice Write the given function as a product of linear and irreducible quadratic factors, each with real coefficients.

Whiteboard Practice Write the given function as a product of linear and irreducible quadratic factors, each with real coefficients.