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