Synthetic Substitution

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

Formula

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.

Example

Example

Solution

Instead of putting 7 into the function,
do the synthetic substitution.

f(x) = x4 - 9x3 + 15x2 + 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.