M3U4D3 Warm Up• Without a

calculator, divide the following

113277323

Solution: 49251

NEW SEATS

Homework Check:

Homework Check:

Homework Check:

Homework Check:

(3a+8b)(3a-8b)

M3U3D3 Synthetic Division

OBJ: To solve polynomial equations

involving division.

Synthetic Division -

4 26 : 5 4 6 ( 3)Ex x x x x

To use synthetic division:

•There must be a coefficient for every possible power of the variable.

•The divisor must have a leading coefficient of 1.

divide a polynomial by a polynomial

1

SWC to Demonstrate using long division first!

Step #1: Write the terms of the polynomial so the

degrees are in descending order.

5x4 0x3 4x2 x 6Since the numerator does not contain all the powers of x, you must include a 0 for thex3.

4 2

5 4 6 ( 3)x x x x

Step #2: Write the constant r of the divisor x-r to the left and write down the coefficients.

Since the divisor is x-3, r=3

5x4 0x3 4x2 x 6

5 0 -4 1 63

4 25 4 6 ( 3)x x x x

5

Step #3: Bring down the first coefficient, 5.

3 5 0 - 4 1 6

4 25 4 6 ( 3) x x x x

5

3 5 0 - 4 1 6

Step #4: Multiply the first coefficient by r, so 3 5 15

and place under the second coefficient then add.

15

15

4 2

5 4 6 ( 3)x x x x

5

3 5 0 - 4 1 6

15

15

Step #5: Repeat process multiplying the sum, 15, by r;

and place this number under the next coefficient, then add.

15 3 45

45

41

4 2

5 4 6 ( 3)x x x x

5

3 5 0 - 4 1 6

15

15 45

41

Step #5 cont.: Repeat the same procedure.

123

124

372

378

Where did 123 and 372 come from?

4 2

5 4 6 ( 3)x x x x

Step #6: Write the quotient.The numbers along the bottom are coefficients of the power of x in descending order, starting with the power that is one less than that of the dividend.

5

3 5 0 - 4 1 6

15

15 45

41

123

124

372

378

4 2

5 4 6 ( 3)x x x x

The quotient is:

5x3 15x2 41x 124 378

x 3

Remember to place the remainder over the divisor!

4 2

5 4 6 ( 3)x x x x

5x 5 21x4 3x3 4x2 2x 2 x 4 Ex 2:

Step#1: Powers are all accounted for and in descending order.

Step#2: Identify r in the divisor.

Since the divisor is x+4, r=-4 .

4 5 21 3 4 2 2

SWC to Demonstrate using long division first!

Step#3: Bring down the 1st coefficient.

Step#4: Multiply and add.

4 5 21 3 4 2 2

-5

Step#5: Repeat.

20 4 -4 0 8-1 1 0 -2 10

4 3 2 105 2

4x x x

x

5 4 3 25 21 3 4 2 2 4x x x x x x

Try this one:

3 21) ( 6 1) ( 2)t t t

2 311 8 16

2Quo i tt t

tent

2 1 6 0 1

2 16 32

1 8 16 31

SYNTHETIC DIVISION:SYNTHETIC DIVISION: Practice PracticeSYNTHETIC DIVISION:SYNTHETIC DIVISION: Practice Practice[1]

[2] [3]

3

1144014623 234

xxxxx

1144014623

120421869

6240733

1245 )3)(62473( xxxxx

)4()1252( 23 xxxx

)1()10135( 234 xxxx

-4 1 2 -5 12-4 8 -12

1 -2 3 0

-1 1 -5 -13 0 10 -1 6 7 -7

1 -6 -7 7 3

322 xx

1

3776 23

x

xxx

The Remainder TheoremThe remainder obtained in the synthetic division process

has an important interpretation, as described in the Remainder Theorem.

The Remainder Theorem tells you that synthetic division can be used to evaluate a polynomial function. That is, to evaluate a polynomial function f (x) when x = k, divide f (x) by x – k. The remainder will be f (k).

Example Using the Remainder Theorem

Use the Remainder Theorem to evaluate the following function at x = –2. f (x) = 3x3 + 8x2 + 5x – 7

Solution:• Using synthetic division, you obtain the following.

Example – SolutionBecause the remainder is r = –9, you can conclude that f (–2) = –9.

This means that (–2, –9) is a point on the graph of f. You can check this by substituting x = –2 in the original function.

Check: f (–2) = 3(–2)3 + 8(–2)2 + 5(–2) – 7 = 3(–8) + 8(4) – 10 – 7 = –9

r = f (k)

cont’d

REMAINDER THEOREM:

Given a polynomial function f(x): then f(a) equals the remainder of

Example: Find the given value

)3(find , 385)( 24 fxxxxf

2 1 3 - 4 -7 2 10 12

1 5 6 5

)(

)(

ax

xf

Method #1: Synthetic Division Method #2: Substitution/ Evaluate

5)2(

78128)2(

7)2(4)2(3)2()2( 23

f

f

f

[A] )2(find , 743)( 23 fxxxxf

[B]

9 3244581)3(

3)3(8)3(5)3()3( 24

f

f-3 1 0 - 5 8 -3

-3 9 -12 12

1 -3 4 -4 9

Classwork

1. Algebra 2 Notes: Synthetic Division

Document Camera

Classwork2. M3U3D3 Using the Remainder Theorem

to Prove Zeroes

Document Camera

HomeworkU3D3

Synthetic Division and the Remainder Theorem.