Chapter 6

Section 3

Dividing Polynomials

Long Division Vocabulary Reminders

quotient

dividenddivisor

Remember Long DivisionRemember Long Division1. Does 8 go into 6?

• No

2. Does 8 go into 64?• Yes, write the integer on

top.

3. Multiply 8∙8• Write under the dividend

4. Subtract and Carry Down5. How many times does 8

go into 7 evenly?• 0 write over the 7

6. Multiply 0∙8 7. Subtract and write

remainder as a fraction.

6478

The divisor and quotient are only FACTORS if the remainder is Zero.

quotient

dividenddivisor

Examples with variables

22xx 393 yy

2164 ww 3362 xx

Examples If the divisor has more than one term,

always use the term with the highest degree.

A remainder occurs when the degree of the dividend is less than the degree of the divisor

Example:

7621 2 xxx

Try These ExamplesDivide using long division.

)1()783( 2 xxx

)3()146( 2 xxx

Long division of polynomials is tedious!

Lets learn a simplified process!This process is called Synthetic Division

p. 316It may look complicated, but watch a few examples and you will get the hang of it.

Use synthetic division to divide 3x3-4x2+2x-1 by x+1

1. Reverse the sign of the constant term in the divisor.Write the coefficients of the polynomial in standard form (Remember to include zeros) Translation: Instead of

write2. Bring down the first coefficient3. Multiply the first coefficient by the new

divisor. Add.4. Repeat step 3 until the end. The last

number is the remainder.5. NOW write the polynomial.

To write the answer use one less degree than the original polynomial.

12431 2 xxxx

-1 3 -4 2 -1

Example: Use synthetic division to divide

a) x3+4x2+x-6 by x+1

b) x3-2x2-5x+6 by x+2

Remainder Theorem

If a polynomial is being divided by (x-a) then the remainder is P(a).

Example: Use the remainder theorem to find P(-4) for P(x)=x3-5x2+4x+12

DO NOT change the number P(a) to -a

Try This Problem

Use synthetic division to find P(-1) for P(x)=4x4+6x3-5x2-60

Homework

Practice 6.3 Evens