Synthetic Division

29 October 2010

Operations on Polynomials

Recap – We know how to: Add Subtract Multiply

What about division?

)2()11183( 24 xxxx

Dividing Polynomials

Like long division We have a short cut!

Synthetic Division!!!

Dividing Polynomials

Synthetic Division Only works when we divide by 1st degree (linear)

polynomials

)2()11183( 24 xxxx

My degree can’t be larger than 1!

Synthetic Division

)24()1152( 24 xxxx

)2()11183( 24 xxxx

Your Turn

On the “Synthetic Division” handout, complete problems 1 – 5. You will: Decide if it’s possible to use synthetic division to

divide the two polynomials

Division Vocab Review

Dividend Divisor

2)3()65( 2 xxxx

Quotient

Preparing for Synthetic Division

Can only be used when the divisor is in the form

x – c

If the divisor isn’t in the form x – c, then you need to convert the expression to include subtraction.

Preparing for Synthetic Division, cont.

5x 11x)11(11 xx

Preparing for Synthetic Division, cont.

Polynomials need to be written in expanded, standard polynomial form. Translation: If you’re missing terms, then you

need to write them out as 0 times (*) the appropriate term.

Preparing for Synthetic Division, cont.

xxx 273 35

xxx 273 35

020703 2345 xxxxx

Your Turn

On “Synthetic Division” handout, write the dividend in expanded standard polynomial form for problems 6 – 10.

Write the divisor in the form x – c.

))2(()0208(

)2()28(23

3

xxxx

xxx

*Synthetic Division Steps

Example Problem:

)2()11183( 24 xxxx

Prep Step

Divisor x – c? x – 2

Dividend in Expanded Standard Polynomial Form? 3x4 – 8x2 – 11x + 1 3x4 + – 8x2 – 11x + 1 3x4 + 0x3 – 8x2 – 11x + 1

Step 1

2

Write the constant value of the divisor (c) here.

Step 2

2

Write all the coefficients of the expanded dividend here.

3 0 -8 -11 1

Step 3

2

“Drop” the 1st coefficient underneath the line.

3 0 -8 -11 1

3

Step 4

2

Multiply “c” by the last value underneath the line. Write their product just underneath the next coefficient.

3 0 -8 -11 1

6

3

Step 5

2

Add together the numbers in that column and write their sum underneath the line.

3 0 -8 -11 1

6

3 6

Step 6

2

Multiply “c” by the last value underneath the line. Write their product just underneath the next coefficient.

3 0 -8 -11 1

6 12

3 6

Step 7

2

Repeat steps 5 and 6 until a number appears in the box underneath the last column.

3 0 -8 -11 1

6 12 8 -6

3 6 4 -3 -5

Step 8 – Naming the Quotient

2

In the last row are the coefficients of the quotient in decreasing order. The quotient is one degree less than the dividend.

3 0 -8 -11 1

6 12 8 -6

3 6 4 -3 -5

Step 8 – Naming the Quotient

3 6 4 -3 -5

The number in the box is the remainder.

)2()11183( 24 xxxx3x3 + 6x2 + 4x – 3 Remainder -5

Your Turn

On the “Synthetic Division” handout, solve for the quotient of problems 11 – 14 using synthetic division.

You may work with your partner.