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).

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