IBHM 086 107.pdf


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4 Polynomials

The same is true for algebraic division. Synthetic division is a shortcut for dividing
polynomials by linear expressions – algebraic long division is covered later in the chapter.

Synthetic division works
only for linear divisors.

Synthetic division works in exactly the same way as the nested calculation scheme. The
value of x that is used is the root that the divisor provides. This is best demonstrated by
example.

Example
Divide 3x3 ⫺ x2 ⫹ 2x ⫺ 5 by x ⫺ 2 using synthetic division.
We need the value of x such that x ⫺ 2 ⫽ 0, that is, x ⫽ 2.
2

⫺1

2

⫺5


6


10


24

3

5

12

19

2

x

3

x

These numbers are the coefficients of the
quotient.

This is the remainder.

So 3x3 ⫺ x2 ⫹ 2x ⫺ 5 ⫽ 1x ⫺ 22 13x2 ⫹ 5x ⫹ 122 ⫹ 19
This could be checked by expanding the brackets.

Example
Divide x3 ⫺ 11x ⫹ 3 by x ⫹ 5.
⫺5

1

1

0

⫺11

3


⫺5


25


⫺70

⫺5

14

⫺67

So x3 ⫺ 11x ⫹ 3 ⫽ 1x ⫹ 52 1x2 ⫺ 5x ⫹ 142 ⫺ 67

Example
Divide 2x3 ⫹ x2 ⫹ 5x ⫺ 1 by 2x ⫺ 1.
Here the coefficient of x in the divisor is not 1.
2x ⫺ 1 ⫽ 0
1
1 2¢x ⫺ ≤ ⫽ 0
2
1
1x⫽
2

89