Complex Plugging

Algebra Level 4

Suppose $f(x)=x^4-8x^3+4x^2+4x+39$ and $f(3+2i)=a+bi$ .

Find the value of $-32\dfrac{a}{b}$ .

Details and Assumptions

Here $i$ is the imaginary unit, which satisfies $i^2=-1$ .

The answer is 4.

2 solutions

Rajen Kapur
Nov 3, 2014

Let x^2 - 6x + 13 having a root 3 + 2i divide f(x) to get a remainder 24( -4x + 13i) which needs plugging to get the answer 4.

There's a typo: It should be $24(-4x + 13)$ instead (without the $i$ ). :) But nice solution!

Happy Melodies - 6 years, 7 months ago

Daniel Carter
Nov 9, 2014

I did it the long way:

f(3+2i) = (3+2i)^4 - 8(3+2i)^3 + 4(3+2i)^2 + 4(3+2i) + 39 = (-119+120i) - 8(-9+46i) + 4(5+12i) + 4(3+2i) + 39 = (-119+120i) + (72-368i) + (20+48i) + (12+8i) +39 = 24-192i

So a = 24 and b = -192

Therefore, the solution we're looking for is:

(-32)*(24/-192) = 768/192 = 4

Complex Plugging

The answer is 4.

2 solutions

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