3.3 Real Zeros of Polynomials

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Presentation transcript:

3.3 Real Zeros of Polynomials

Rational Zero Theorem If the polynomial P(x) has integer coefficients then every rational zero of P is of the form p/q where p is a factor of the constant and q is a factor of the leading coefficient. 3 lists: Factors of p: Factors of q: Possible roots: p/q

List all possible rational roots. Possible roots: p/q P(x) = 9x6 – 5x3 + 27 Factors of p: Factors of q: Final Possibilities Cancel out duplicates

Use the rational zero theorem to factor: P(x) = 3x3 – 4x2 – 13x - 6 Final Possibilities Possible roots: p/q Use synthetic division to find one of the factors This means the factors are: Now factor the quadratic.

Use the rational zero theorem to factor: Final Possibilities P(x) = -x3 – 3x2 + 13x + 15 Use synthetic division to check This means the factors are: Now factor the quadratic.

Find all the real zeros Final Possibilities P(x) = x3 – 6x + 4 Zeros: Use synthetic division to check Zeros: Factors: Use the quadratic formula.

Descartes Rule In some cases this rule is helpful in eliminating candidates from a lengthy list of possible rational roots.

Use Descartes Rule to determine the possible number of positive and negative real zeros of the polynomial. P(x) = x3 + 4x2 + 3x - 2 Only one change in sign Therefore: only one positive real zero Two changes in sign Therefore: only two possible negative real zero

True or False: all zeros of the polynomial P(x) = 4x3 + 19x2 + 11x – 7 lie between -3 and 1? All positive Therefore 1 is an upper bound All alternate between + and -? Therefore -3 is NOT a lower bound

Find all the zeros: P(x) = 2x5 + 5x4 – 8x3 – 14x2 + 6x + 9 Final Possibilities List of Zeros

Find all the zeros: y = 6x4 – 5x3 – 38x2 – 5x + 6 Final Possibilities List of Zeros

Find all the zeros: P(x) = 2x5 - 11x4 – 10x3 + 56x2 + 88x + 35

Homework pg 279 #7, 9, 12, 16, 18, 22, 24, 28, 30-33