Factoring with Factoring with Real Number Real Number CoefficientsCoefficients

Sec. 2.6cHomework: p.235 37-41 odd, 45-49 odd

Factors of a Polynomial with Factors of a Polynomial with Real CoefficientsReal Coefficients

Definition: A quadratic with no real zerosis irreducible over the reals.

Every polynomial function with real coefficientscan be written as a product of linear factorsand irreducible quadratic factors, each withreal coefficients.

The New Rule:

A Practice ProblemA Practice ProblemWrite the given function as a product of linear andirreducible quadratic factors, each with real coefficients.

5 4 3 23 2 6 4 24 16f x x x x x x Graph to find a rational zero.

4 223 6 24

3f x x x x

x = 2/3

2/3 3 –2 6 –4 –24 16

2 0 4 0 –16

3 0 6 0 –24 0

A Practice ProblemA Practice ProblemWrite the given function as a product of linear andirreducible quadratic factors, each with real coefficients.

5 4 3 23 2 6 4 24 16f x x x x x x

4 223 6 24

3f x x x x

4 223 2 8

3f x x x x

How do we factor

this last term?

2 23 2 2 4f x x x x

23 2 2 2 4f x x x x x

A Practice ProblemA Practice ProblemWrite the given function as a product of linear andirreducible quadratic factors, each with real coefficients.

5 4 3 23 2 6 4 24 16f x x x x x x

The zeros of the last term are complex…

23 2 2 2 4f x x x x x

We have taken the factorization of We have taken the factorization of ff as far as as far aspossible, subject to the condition that possible, subject to the condition that eacheachfactor has real coefficients.factor has real coefficients.

Polynomial Function of Polynomial Function of Odd DegreeOdd Degree

A function’s complex zeros always occur inconjugate pairs…

Since a function with odd degree has an oddnumber of zeros, what can we conclude?...

Every polynomial function of odddegree with real coefficients has atleast one real zero.

Does this hold with our previous example???

More Guided PracticeMore Guided PracticeWrite the given function as a product of linear andirreducible quadratic factors, each with real coefficients.

3 2 6f x x x x

22 3f x x x x

Whiteboard PracticeWhiteboard PracticeWrite the given function as a product of linear andirreducible quadratic factors, each with real coefficients.

3 23 2 2f x x x x

21 3 2f x x x x

Whiteboard PracticeWhiteboard PracticeWrite the given function as a product of linear andirreducible quadratic factors, each with real coefficients.

4 3 22 8 12f x x x x x

23 1 4f x x x x