6-3: Dividing Polynomials

6-3: DIVIDING POLYNOMIALSEssential Question: What does the last number in the bottom line of synthetic division represent?

6-3: DIVIDING POLYNOMIALS

Using Long Division Polynomial Long division works similarly to

regular long division (I know… it’s been a while) Divide 1,732,042 by 440

3936440 1732042

1320

4120

3960

1604

1320

2842

2640

202

Answer is 3936, R: 202


Divide x2 + 3x – 12 by x – 3

2

2

6

6 1

3 3 12

6 12

6

8

3

x

x x

x x x

x

x

Answer is x + 6, R: 6


YOUR TURN Divide x2 – 3x + 1 by x – 4

2

2

4 3 1

1

4

1

5

4

x

x

x

x x

x

x

x

Answer is x + 1, R: 5

6-3: DIVIDING POLYNOMIALS We can use division to find factors of a

polynomial. If the remainder of division comes out to be 0, then

the divisor (and quotient) are factors. Determine whether x + 4 is a factor of the

polynomial x2 + 6x + 8.

Because the remainder is 0, x + 4 IS a factor of x2 + 6x + 8 (so is x + 2)

2

2 4

4 6 8

2

2

0

2 8

8

x x

x

x

x

x

x

x


Determine whether x + 4 is a factor of the polynomial x3 + 3x2 – 6x – 7.

Because the remainder is not 0, x + 4 IS NOT a factor of x3 + 3x2 – 6x – 7

3 2

2

2

2

3 2

4 3 6 7

6

4

2

2

1

8

4

2

7

x x

x x x x

x

x x

x x

x

x

x

6-3: DIVIDING POLYNOMIALS YOUR TURN Determine whether x – 8 is a factor of the

polynomial 2x2 – 19x + 24.

Because the remainder is 0, x – 8 IS a factor of 2x2 – 19x + 24

2

2

2

2 16

8 2 19 24

3 24

0

3

3 24

x x x

x

x

x

x

x

6-3: DIVIDING POLYNOMIALS YOUR TURN Determine whether x + 2 is a factor of the

polynomial x3 – 4x2 + 3x +2.

Because the remainder is not 0, x + 2 IS NOT a factor of x3 – 4x2 + 3x + 2

2

2

3

2

3

2

2

6

6 12

15

15 30

2 4 3 2

6 3

2

15 2

28

x x

x x x x

x

x

x

x

x

x

x

x


Assignment Page 324 Problems 1 – 12 (all problems) Obviously, show your work

Tomorrow Quiz review

Next week The secrets of synthetic division Chapter 6 Test

UNIT #4: POLYNOMIALS6-3: DIVIDING POLYNOMIALSDAY 2Essential Question: What does the last number in the bottom line of synthetic division represent?

6-3: DIVIDING POLYNOMIALS Synthetic Division

Can only be used when the divisor is “x” +/- some constant (e.g. “x + 2”, “x – 10”)

Step 1: Reverse the sign of the constant terms in the divisor. Write the coefficients of the polynomial in standard form.

Step 2: Bring down the first coefficient Step 3: Multiply the first coefficient by the new

divisor. Write the result under the next coefficient. Add.

Step 4: Repeat the steps of multiplying and adding until the remainder is found.

Step 5: The quotient begins one degree less than the dividend.


Divide 3x3 – 4x2 + 2x – 1 by x + 13 21 3 4 2 1x x x x

1 3 4 2 1



1 3 4 2 1

3



1 3 4 2 1

3

3 7



1 3 4 2 1

3 7

3 7 9


Divide 3x3 – 4x2 + 2x – 1 by x + 1

The quotient is 3x2 – 7x + 9, R: -10

3 21 3 4 2 1x x x x

1 3 4 2 1

3 7 9

3 7 9 10


YOUR TURN Divide x3 – 4x2 + x – 6 by x – 3

Answer is x2 – 1x – 2, R: -12

3 1 4 1 6

3 3 6

1 1 2 12


Assignment Page 324 – 325 Problems 13 – 22 & 48 – 51 (all problems) Obviously, show your work

Tomorrow Chapter 6 Review Packet will be distributed

6-3: Dividing Polynomials

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