6.3 Synthetic Division

Here is a quick way to evaluate the value of a function for a particular value of x. It is called

synthetic substitution.

Evaluate for x = 3 5425 23 xxxy

54253

5

15

13

39

43

129

124

The value of the function is 124 when x = 3

Use synthetic substitution to find the value of y when x = 2

544535 234 xxxxy

54453512

The value of the function is -100 when x = 2

Use synthetic substitution to find the value of y when x = -2

544535 234 xxxxy

54453512

The value of the function is 0 when x = -2

This means that x = -2 is a root of the function.

Since x = -2 is a root of the function, we know that….

544535))(2( 234 xxxxsomethingx

Divide x4 + 5x3 –3x2 +45x –54 by x + 2 to determine what the something in the parentheses is:

5445352 234 xxxxx

Divide

5445352 234 xxxxx

3x

34 2xx

544533 23 xxx

23x

23 63 xx 54459 2 xx

x9

xx 189 2 5427 x

27

5427 x0

Now compare this result to what you got when you did synthetic substitution. What do you notice?

5445352 234 xxxxx

3x

34 2xx

544533 23 xxx

23x

23 63 xx 54459 2 xx

x9

xx 189 2 5427 x

27

5427 x0

541862

54453512

1 3 -9 -27 0

Synthetic Substitution is the same as Long Division!!!!

(Except that it only works for linear factors)

Compare

Conclusion:

We can use synthetic division in most cases instead of long division.

Since synthetic substitution, synthetic division, and long division are essentially the same process….

Divide:

27128 2 xxx

271218

8

54827122

x

xxxx

Divide:

362123 23456 xxxxxxx

316212113

124

336212245

23456

xxx

xxxxxxx

Since the remainder is zero (x+3) is a factor.

Divide:

5.22345.129 23 xxxx

5.22345.1219

5.25.3

95.22345.122

23

xx

xxxx

Since the remainder is zero (x - 9) is a factor.

What would happen if your divided repeatedly?

84104 2345 xxxxxy

This polynomial has roots at x = 1 and –2 (and no others)

What would happen if your divided repeatedly?

84104 2345 xxxxxy

This polynomial has roots at x = 1 and –2 (and no others)

32 )2()1( xxy

Factor completely using synthetic division.

121014102 234 xxxxy

This polynomial has roots at x = 2 and 3 (and no others)

Factor completely using synthetic division.

121014102 234 xxxxy

This polynomial has roots at x = 2 and 3 (and no others)

)22)(3)(2( 2 xxxy