Let p(x)=2x^3-8x^2-23x+63.  Use synthetic division to determine: p(-2i)

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embizze eNotes educator| Certified Educator

Given `p(x)=2x^3-8x^2-23x+63` find `p(-2i)` using synthetic division.

p(-2i) is the remainder found after using synthetic division. (For this reason, synthetic division is often called synthetic substitution.)

-2i  |   2          -8          -23           63
                       -4i       -8+16i       32+62i
       ---------------------------------------
          2    -8-41          -31+16i     95+62i

Thus p(-2i)=95+62i.

You can check using substitution:

`p(-2i)=2(-2i)^3-8(-2i)^2-23(-2i)+63`

`=2(-8i^3)-8(4i^2)-23(-2i)+63`

`=-16i^3-32i^2+46i+63`

`=16i+32+46i+63`

`=95+62i`

justaguide eNotes educator| Certified Educator

The polynomial p(x)=2x^3-8x^2-23x+63

p(-2i) = 2*(-2i)^3-8(-2i)^2-23*(-2i)+63

= 2*-8*i^3-8*4*i^2+ 46i+63

= 2*8*i+8*4+ 46i+63

= 16i + 32 + 46i + 63

= 62i + 95

The polynomial p(-2i) = 62i + 95

sunwers | Student

would this be considered as using synthetic division by replacing x with -2i?

sunwers | Student

sorry that -65 should have been a -2 as the remainder I typed it in wrong.

sunwers | Student

I'm confused on this one... thanks justaguide but it says to use synthetic division to determine.  I first thought that you would replace x with "-2i" and I did go about solving this problem the way you said above.  But all the rest of my synthetic divsion in this chapter dealt with using it like the following

Sorry trying to type a problem on a forum is hard for me.  5 being in the box and the 2 brought straight down.

5      2 -  8 - 23 + 63

          +10 +10 -  65

______________________

        2 + 2 - 13 - 65

Answer being:  2x^2+2x-13+ 2/x-5