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.) FactorA)

B)

Number System

REAL IMAGINARYRATIONAL IRRATIONAL iend or don’t end a+biRepeat don’t repeatIntegers square rootsFractions π or e

Rational Zeros Test

• The POSSIBLE rational 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.