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So I have to find the value of : $$2x^3+2x^2-7x+72$$ at $$x= \frac{3-5i}{2} $$ Where $i$ stands for $\sqrt{-1}$. I know the obvious approach would be to substitute the value of $x$ in the equation, but the equation gets extremely messy in that process. Is there any easier or quicker approach for the sum.

P.s. The answer is $4$.

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    Do synthetic division. – 2017-01-20
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    @Masacroso sorry. My bad – 2017-01-20
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    @AhmedS.Attaalla Please explain it. I have done it for other questions, but I can't understand how to apply it in the case of a complex polynomial – 2017-01-20
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    Write it as $72 - x\big( 7 - 2x(1+x)\big)$ and work from the inside out. – 2017-01-20

3 Answers 3

5

You have $\frac{3-5i}{2}$ is the solution of $2x^2-6x+17 = 0$.

Then, $$2x^3 + 2x^2-7x+72 = x(2x^2-6x+17) + 4(2x^2-6x+17) + 4 = 4.$$

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Hint

$$x= \frac{3-5i}{2}$$ $$x^2=x \times x=-4-\frac{15 i}{2}$$ $$x^3=x \times x^2=-\frac{99}{4}-\frac{5 i}{4}$$

1

By the polynomial remainder theorem $f(\frac{3-5i}{2})$ is simply the remainder we get when we divide $f(x)$ by $x-\frac{3-5i}{2}$. So perform synthetic division, which I think is easier.