How do you use synthetic substitution to find P(2) for # P(x) = 4x^3 - 5x^2 + 7x - 9#?

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Answer 1 · 2015-06-19T22:52:27

#P(2)# is the remainder when #P(x)# is divided by #x-2#. You can use synthetic division to do that division. (You could also use long division.)

I'm still working on finding a good way to format division here, but if you know synthetic division at all, I think this will get the idea across.

#{: (1, "|", 4, -5, 7, -9),(color(white)"1", "|", color(white)"ss", 8, 6, 26), (color(white)"1", color(white)"1", 4, 3, 13, 17):}#

The quotient is #4x^2+3x+13# and the remainder (which is what we want) is #17#.

So, the Remainder Theorem tells us that #P(2) = 17#.

It may not seem like a big deal, but for many people this is faster tan evaluating #P(2)# by doing:

#4(2)^3-5(2)^2+7(2)-9# to get #17#

And there are other uses of this fact (theorem).