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5.6A Rational Zeros Theorem. BELL RINGERS 1.) Which are RATIONAL numbers? A) -5 B) ¾ C) D)-.5 E) 4i F) 2/3 G) .333… 2. ) Divide with synthetic division. ( (x – 4) 3.) Factor A) B) . Number System. REAL IMAGINARY RATIONAL IRRATIONAL i

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5.6A Rational Zeros Theorem • BELL RINGERS • 1.) Which are RATIONAL numbers? A) -5 B) ¾ C) D)-.5 E) 4i F) 2/3 G) .333… • 2. ) Divide with synthetic division. ( (x – 4) • 3.) Factor A) B)

Number System REALIMAGINARY RATIONALIRRATIONAL i end or don’t end a+bi Repeat don’t repeat Integers square roots Fractions π or e

Rational Zeros Test • The POSSIBLErational zeros (solutions) of a polynomial function are given by • = • Factors= integers that can be multiplied to create the number. • Put polynomial in STANDARD FORM first • All factors can be

Examples: List the POSSIBLE Rational Zeros • 1. • 2. f(x) =

Examples: List POSSIBLE Rational Zeros • 3. • 4. f(x) =

Finding ALL REAL Solutions: • A graph can shorten your list. • Focus on #’s in list that appear to be where the graph touches x-axis (x-intercepts) • Synthetically divide by a GOOD zero from graph. Remainder = ZERO if it is a solution • If new quotient is degree of 2: factor & solve or quadratic formula to solve • If new quotient is degree of 3 or more, synthetically divide NEW quotient by another GOOD zero from graph, & solve. • IF no graph given: choose #’s from list & synthetically divide until Remainder = zero, then synthetically divide NEW quotient by more #’s from your list until you can factor & solve to find ALL solutions (or quadratic formula)

Examples: Shorten list & find ALL real zeros • 5. X-intercepts in graph: -3,# between -1&-2, # between 0 & 1.

Example: Shorten list and find ALL Real Zeros • 6. X-Intercepts in graph: 1, # between 1 & 2, # between -2 & -3.