[PDF] 23 Synthetic Division 23 Remainder Theorem 23 Factor Theorem

101361_6Week8PreCalcHonors.pdf Name: ____________________________________________________________ Period: ________

Pre-Calculus Honors

Chapter 2: Polynomial & Rational Functions

Monday

October 4

2.3 Synthetic Division

Tuesday

October 5

2.3 Remainder Theorem

Wednesday

October 6

2.3 Factor Theorem

Thursday

October 7

2.3 Rational Zeros

Friday

October 8

2.3 Rational Root Theorem

2.3 Real Zeros of Polynomial Functions: Synthetic Division

1

Synthetic Division

When dividing polynomials, there is a shortcut method that you can use called synthetic division. To divide ܽݔଷ൅ܾݔଶ൅ܿ

vertical pattern is to add terms; the diagonal pattern ŝƐƚŽŵƵůƚŝƉůLJďLJ͞Ŭ͟ĂƐLJŽƵǁŝůůƐĞĞďĞůŽǁ͘

Example 1: Solve with synthetic division. ሺݔସെͳͲݔଶെʹݔ൅Ͷሻൊሺݔ൅͵ሻ

Example 2: Solve with synthetic division. ሺͷݔଷ൅ͺݔଶെݔ൅͸ሻൊሺݔെʹሻ

Example 3: Solve with synthetic division. ሺʹݔଷ൅ͳ͵ݔെͳͲሻൊሺݔ൅Ͷሻ

2.3 Real Zeros of Polynomial Functions: Remainder Theorem

1

Remainder Theorem

The Remainder Theorem ƐĂLJƐ͕͞ĨĂƉŽůLJŶŽŵŝĂů݂ሺݔሻ is divided by ሺݔെ݇ሻ, then the remainder

is ݂ሺ݇ሻ.

Example 1: Use the Remainder Theorem to evaluate ݂ሺݔሻൌ͵ݔଷ൅ͺݔଶ൅ͷݔെ͹

when ݔൌെʹǤ Example 2: Use the Remainder Theorem to find each function value given: ݂ሺݔሻൌͶݔଷ൅ͳͲݔଶെ͵ݔെͺ Find: ݂ሺെͳሻǡ݂ሺͶሻǡ݂ቀଵ ଶቁǡܽ

2.3 The Rational Zero Test

The Rational Zero Test (or Rational Root Theorem)

The Rational Zero Test relates the possible rational zeros of a polynomial to the leading coefficient and

the constant term of the polynomial.

ŽƌƚŚŝƐ͞ƚŚĞŽƌĞŵ͕͟ǁĞƵƐƵĂůůLJƌĞĨĞƌƚŽƚŚĞůĞĂĚŝŶŐĐŽĞĨĨŝĐŝĞŶƚĂƐ͞Ƌ͟ĂŶĚƚŚĞĐŽŶƐƚĂŶƚƚĞƌŵĂƐ͞Ɖ͘͟ŚĞ

possible rational roots of a polynomial function are ௧௛௘௙௔௖௧௢௥௦௢௙௣

௧௛௘௙௔௖௧௢௥௦௢௙௤ or ௧௛௘௙௔௖௧௢௥௦௢௙௧௛௘௖௢௡௦௧௔௡௧௧௘௥௠

௧௛௘௙௔௖௧௢௥௦௢௙௧௛௘௟௘௔ௗ௜௡௚௖௢௘௙௙௜௖௜௘௡௧.

Once you find the possible rational roots, you test them in your equation (using synthetic division) to see

which of the possible roots are actually roots of the equation. Usually, you use this method for polynomial equations that you cannot factor.

Example 1: Find the possible rational roots (I will abbreviate this as prr from now on) of the equation

ݕൌݔଷെͷݔଶ൅ʹݔ൅ͺǤ

Žǁ͕ƵƐĞƐLJŶƚŚĞƚŝĐĚŝǀŝƐŝŽŶƚŽ͞ƚĞƐƚ͟ǁŚŝĐŚŽĨƚŚĞƐĞprr is a real root of the equation. (Note: If you have

a cubic polynomial, you need to find one prr ƚŚĂƚ͞ǁŽƌŬƐ͟ďĞĨŽƌĞLJŽƵĐĂŶĨĂĐƚŽƌŽƌƵƐĞƋƵĂĚƌĂƚŝĐ

formula on the remaining quadratic polynomial; for a quartic polynomial, you need to find two prr and

for a quinƚŝĐ͕LJŽƵ͛ůůŶĞĞĚƚŽĨŝŶĚƚŚƌĞĞprr).

Now, factor completely and solve for the real zeros of the equation.

2.3 The Rational Zero Test

Example 2: Find the real zeroes of ݂ሺݔሻൌ͸ݔଷെͶݔଶ൅͵ݔെʹ.

Example 3: Find the real zeros of ݂ሺݔሻൌͺݔଷെͶݔଶ൅͸ݔെ͵Ǥ

Example 4: Find the real zeros of ݂ሺݔሻൌͳͲݔସെͳͷݔଷെͳ͸ݔଶ൅ͳʹݔ.