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

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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:

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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`

Approved by eNotes Editorial Team
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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

Approved by eNotes Editorial Team