Synthetic Substitution

How to use the synthetic substitution to find the value of a function: formula, examples, and their solutions.

Formula

Recall that
the remainder of f(x) ÷ (x - a) is f(a).

Remainder theorem

And by using synthetic divition,
you can find the remainder of f(x) ÷ (x - a), f(a).

Synthetic division

So, to find f(a),
you can find the remainder of f(x) ÷ (x - a)
by using synthetic division.

This process is called
[synthetic substitution].

Example 1: f(x) = x4 - 9x3 + 15x2 + 3x - 62, f(7) = ?

Instead of putting 7 into f(x),
use the synthetic division
to find f(7).

Write the coefficients of the terms of f(x):
1 -9 15 3 -62.

Write the L shape form like this.

And write 7
next to the form.

Synthetic division

Do the synthetic division.

: 1
↗: 1⋅7 = 7
: -9 + 7 = -2
↗: -2⋅7 = -14
: 15 - 14 = 1
↗: 1⋅7 = 7
: 3 + 7 = 10
↗: 10⋅7 = -70
: -62 + 70 = -8

The remainder is -8.

So, by the remainder theorem,
f(7) = -8.

Remainder theorem

Example 2: f(x) = x5 - 4x4 + 5x3 - 20x + 7, f(3) = ?

Instead of putting 3 into f(x),
use the synthetic division
to find f(3).

Write the coefficients of the terms of f(x):
1 -4 5 0 -20 7.

(Don't forget to write the coefficient of x2 term: 0.)

Write the L shape form like this.

And write 3
next to the form.

Do the synthetic division.

: 1
↗: 1⋅3 = 3
: -4 + 3 = -1
↗: -1⋅3 = -3
: 5 - 3 = 2
↗: 2⋅3 = 6
: 0 + 6 = 6
↗: 6⋅3 = 18
: -20 + 18 = -2
↗: -2⋅3 = -6
: 7 - 6 = 1

The remainder is 1.

So, by the remainder theorem,
f(3) = 1.