Question

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

Answer 1

-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*-3=-12`
- write -12 below the -5.
- add -5 and -12. `-5+(-12)=-17`
- multiply -17 by -3. `-17*-3 = 51`
- write 51 below the 7.
- add 51 and 7. `7+51=58`
- multiply 58 by -3. `58*-3=-174`
- write -174 below the -10.
- add -174 and -10. `-10+(-174)=-184`

i.e. `p(-3)=-184`

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

`4*(-3)^3= 4*9*-3 = 36*-3=-108`
`-5*(-3)^2=-5*9=-45`
`7*(-3)=-21`
`-10=-10`
i.e. `p(-3)=-108-45-21-10=-184`

That's it. Hope it helped.