# Synthetic Substitution

How to do the synthetic substitution to find the function value f(a): formula, 1 example, and its solution.

## Formula

Recall that

the remainder of f(x)/(x - a) is

f(a).

Remainder Theorem

And by doing the synthetic division,

you can find the remainder:

it's the bottom right number

covered by the L shape form.

So

f(a) is the remainder of the synthetic division

of f(x)/(x - a).

This is the synthetic substitution.

## Examplef(x) = x^{4} - 9x^{3} + 15x^{2} + 3x - 62, f(7) = ?

Instead of putting 7 into the function,

do the synthetic substitution.

f(x) = x^{4} - 9x^{3} + 15x^{2} + 3x - 62

Write the coefficients of the terms

in descending order:

1 -9 15 3 -62.

Draw an L shape form like this.

To find f(7),

write 7

on the left side of the form.

Then do the synthetic division.

This is the synthetic division of f(x)/(x - 7).

↓: 1 = 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

Draw another L shape form

that covers the right end number 8.

This is the synthetic division of

f(x)/(x - 7).

This 8 is the remainder of f(x)/(x - 7).

So f(7) = 8.

So 8 is the answer.