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A). B). Polynomial Long Division Review. STEP #1 : Write the Polynomial in DESCENDING ORDER by degree and write any ZERO coefficients for missing degree terms in order. SYNTHETIC DIVISION:. STEP #2 : Solve the Binomial Divisor = Zero.

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synthetic division

STEP #1: Write the Polynomial in DESCENDING ORDER by degree and write any ZERO coefficients for missing degree terms in order

SYNTHETIC DIVISION:

STEP #2: Solve the Binomial Divisor = Zero

STEP #3: Write the ZERO-value, then all the COEFFICIENTS of Polynomial.

Zero = 2

5 -13 10 -8 = Coefficients

STEP #4 (Repeat):

(1) ADD Down, (2) MULTIPLY, (3) Product  Next Column

synthetic division continued

Zero = 2

5 -13 10 -8 = Coefficients

10

-6

8

5

-3

0 = Remainder

4

SYNTHETIC DIVISION: Continued

STEP #5: Last Answer is your REMAINDER

STEP #6: POLYNOMIAL DIVISION QUOTIENT

Write the coefficient “answers” in descending order starting with a Degree ONE LESS THAN Original Degree and include NONZERO REMAINDER OVER DIVISOR at end

(If zero is fraction, then divide coefficients by denominator)

5 -3 4 

SAME ANSWER AS LONG DIVISION!!!!

synthetic division practice

[1]

Zero =

= Coefficients

SYNTHETIC DIVISION: Practice

[2]

[3]

[4]

Divide by 2

slide5

REMAINDER THEOREM:

Given a polynomial function f(x):

then f(a) equals the remainder of

Example: Find the given value

[A]

Method #2: Substitution/ Evaluate

Method #1: Synthetic Division

2 1 3 - 4 -7

2 10 12

1 5 6 5

-3 1 0 - 5 8 -3

-3 9 -12 12

1 -3 4 -4 9

[B]

slide6

FACTOR THEOREM:

(x – a) is a factor of f(x) iff f(a) = 0 remainder = 0

Example: Factor a Polynomial with Factor Theorem

Given a polynomial and one of its factors, find the remaining factors using synthetic division.

-3 1 3 -36 -108

-3 0 108

1 0 -36 0

(Synthetic Division)

(x + 6) (x - 6)

Remaining factors

slide7

PRACTICE: Factor a Polynomial with Factor Theorem

Given a polynomial and one of its factors, find the remaining factors.

[A]

STOP once you have a quadratic!

[B]

STOP once you have a quadratic!

slide8

Finding EXACT ZEROS (ROOTS) of a Polynomial

[1]FACTORwhen possible & Identify zeros:

Set each Factor Equal to Zero

[2a]All Rational Zeros =

P = leading coefficient, Q = Constant of polynomial

[2b] Use SYNTHETIC DIVISION

(repeat until you have a quadratic)

[3]Identify the remaining zeros

 Solve the quadratic = 0

(1) factor (2) quad formula (3) complete the square

Answers must be exact, so factoring and graphing won’t always work!

slide9

Example 1:Find ZEROS/ROOTS of a Polynomial by FACTORING: (1) Factor by Grouping (2) U-Substitution

(3) Difference of Squares, Difference of Cubes, Sum of Cubes

[B]

[A]

Factor by Grouping

Factor by Grouping

[D]

[C]

slide10

Example 2:Find ZEROS/ROOTS of a Polynomial by SYNTHETIC DIVISION (Non-Calculator)

  • Find all values of
  • Check each value by synthetic division

[B]

[A]

Possible Zeros (P/Q)

±1, ±2

Possible Zeros (P/Q)

±1, ±3, ±7, ±21

slide11

Example 2:PRACTICE

[D]

[C]

Possible Zeros (P/Q)

±1, ±3

Possible Zeros (P/Q)

±1, ±2, ±4, ±8

slide12

Example 2:PRACTICE

[F]

[E]

Possible Zeros (P/Q)

±1, ±2, ±4, ±1/2

Possible Zeros (P/Q)

±1, ±2, ±3, ±6, ±1/2, ± 3/2

slide13

Example 2:PRACTICE

[H]

[G]

Possible Zeros (P/Q)

±1, ±2, ±3, ±6, ±1/3, ± 2/3

Possible Zeros (P/Q)

±1, ±2, ±1/2 ±1/3, ±2/3 , ± 1/6

slide14

Example 3:Find ZEROS/ROOTS of a Polynomial by GRAPHING (Calculator)

  • [Y=], Y1 = Polynomial Function and Y2 = 0
  • [2ND] [TRACE: CALC] [5:INTERSECT]
  • First Curve? [ENTER], Second Curve? [ENTER]
  • Guess? Move to a zero [ENTER]

[A]