# 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*) = *x*^{4} - 9*x*^{3} + 15*x*^{2} + 3*x* - 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*) = *x*^{5} - 4*x*^{4} + 5*x*^{3} - 20*x* + 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 *x*^{2} 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.