# How do you use synthetic substitution to find f(3) and f(-2) for f(x)=x^2-5?

Sep 9, 2015

$f \left(3\right) = 4$

#### Explanation:

The process is very close to synthetic division except in this case we are actually looking for the remainder. Of course in this case is much easier for find f(3) and f(-2) by direct substitution, but for higher degree polynomials with more terms the substitution becomes rather cumbersome, so let's try the synthetic substitution:

f(x) = x^2 - 5 , f(3) = ?

Set up:
On the 1st row write the coefficients (in this case 1, 0 and -5, notice that 0 is the coefficient of x). On the 2nd row and enter the value you need to substitute(in this case 3). On the 3rd row copy down the leading coefficient as follow:

$\ldots \ldots . 1 \ldots \ldots \ldots 0 \ldots \ldots - 5$
$3 \ldots . \downarrow$
$\ldots \ldots . 1$

Operation:
Multiply 3 on 2nd row by 1 on 3rd row and write the results on the 2nd row under 0. Add the two values and write the results directly below them on 3rd row in this case we get 3. Now again multiply 3 in the 2nd row by the 3 you just wrote down, and enter the results in this case 9 under -5 then add the 9 and -5 and write the results on the 3rd row right below them, this value which is 4 is your f(3).

$\ldots \ldots . 1 \ldots \ldots \ldots 0 \ldots \ldots - 5$
$3 \ldots . \downarrow$..... ...3 .............9
$\ldots \ldots . 1 \ldots \ldots \ldots 3. . . \ldots \ldots . 4$ =>$f \left(3\right) = 4$

I let you practice with f(-2).