120223 Polynomial function summary

The only rational numbers that can possibly be zeros

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The only rational numbers that can possibly be zeros of P ( x ) are those of the form a k of factors of factors ± . No other rational numbers need to be checked. Complex Conjugate Zeros: For P ( x ) with real coefficients, any non-real zeros occur as “conjugate” pairs: if x = u + vi is a zero, then x = u vi is also a zero. Factoring hints [Methods marked with are still to be covered, on Friday.] Look for common factors, such as having an x in every term. Remember that besides all the factors corresponding to the zeros, a polynomial may need to have a numerical factor. For example: 2 x 2 + 8 x + 6 = 2( x + 1)( x + 3), not just ( x + 1)( x + 3). Look for special forms such as x 2 y 2 , x 3 + y 3 , and x 3 y 3 . See class notes or p. 441 for how to factor these. Try temporarily substituting another variable name for a power of x . For example, if all of the terms have even powers of x , try the substitution y = x 2 . There’s a special technique for cubic-or-higher polynomials where there turn out to be two factors with just two terms each, sometimes called “factoring by grouping.”
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