Synthetic Division

APPENDIX

B

723

OBJECTIVE 1 Use synthetic division to divide by a polynomial of the form

xk.If a polynomial in xis divided by a binomial of the form a shortcut

method can be used. For an illustration, look at the division on the left below.

 x- k,

OBJECTIVES

1 Use synthetic

division to divide by

a polynomial of the

form xk.

2 Use the remainder

theorem to evaluate

a polynomial.

3 Decide whether a

given number is a

solution of an

equation.



80

25 - 75

25 5

9 - 27

9 - 2

3 - 9

1 - 3  30 - 25

3 9 25

80

25 x- 75

25 x+ 5

9 x^2 - 27 x

9 x^2 - 2 x

3 x^3 - 9 x^2

x- 3  3 x^3 + 0 x^2 - 2 x+ 5

3 x^2 + 9 x+ 25

On the right above, exactly the same division is shown written without the vari-

ables. This is why it is essentialto use 0 as a placeholder in synthetic division. All the

numbers in color on the right are repetitions of the numbers directly above them, so

we omit them, as shown on the left below.

80

- 75

25

- 27

9

- 9

1 - 3  30 - 25

3 9 25

80

- 75

25 5

- 27

9 - 2

- 9

1 - 3  30 - 25

3 9 25

The numbers in color on the left are again repetitions of the numbers directly above

them. They too are omitted, as shown on the right above. If we bring the 3 in the div-

idend down to the beginning of the bottom row, the top row can be omitted, since it

duplicates the bottom row.

We omit the 1 at the upper left, since it represents 1x, which will always be the first term

in the divisor. Also, to simplify the arithmetic, we replace subtraction in the second row

by addition. To compensate for this, we change the at the upper left to its additive

inverse, 3.

- 3

3 9 25 80

- 9 - 27 - 75

1 - 3  30 - 25