# How do you use synthetic substitution to find p(-3) for p(x)=4x^3-5x^2+7x-10?

Sep 20, 2015

-184

#### Explanation:

Well, you need to know what a synthetic substitution is. I am not going to explain this here but just show you the steps.
I didn't know what it was until I read this question so I googled it and found some great Youtube videos that explain how to do the trick. Here is one, for example

So in this question, we want to find p(-3) so write down $- 3$ on the left, and the coefficients of the polynomial $4 , - 5 , 7 , - 10$.

Here are the steps:
- drop down the 4.
- multiply 4 by -3. $4 \cdot - 3 = - 12$
- write -12 below the -5.
- add -5 and -12. $- 5 + \left(- 12\right) = - 17$
- multiply -17 by -3. $- 17 \cdot - 3 = 51$
- write 51 below the 7.
- add 51 and 7. $7 + 51 = 58$
- multiply 58 by -3. $58 \cdot - 3 = - 174$
- write -174 below the -10.
- add -174 and -10. $- 10 + \left(- 174\right) = - 184$

i.e. $p \left(- 3\right) = - 184$

You can check if this answer is correct by directly substituting in the original equation:

$4 \cdot {\left(- 3\right)}^{3} = 4 \cdot 9 \cdot - 3 = 36 \cdot - 3 = - 108$
$- 5 \cdot {\left(- 3\right)}^{2} = - 5 \cdot 9 = - 45$
$7 \cdot \left(- 3\right) = - 21$
$- 10 = - 10$
i.e. $p \left(- 3\right) = - 108 - 45 - 21 - 10 = - 184$

That's it. Hope it helped.