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What will be the value of $f(x)=3x^4+14x^3+24x^2-6x-10$, when $x=-2+\sqrt{3}i$? I can evaluate multiplying 4 times. I am looking for a shortcut method?

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    $\begingroup$ Why not just plug this value in the function and calculate the result? $\endgroup$
    – roman
    Dec 18, 2018 at 15:56
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    $\begingroup$ @roman it's not needed. $\endgroup$ Dec 18, 2018 at 16:03

2 Answers 2

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TIP: $(x+2)^2 = 3(-1) \implies x^2 + 4x + 7 = 0$. Divide the polynomial from the question by this one and reduce the problem to expand the linear remainder.

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Hint $\ x^2+4x+7 = 0 $ so by division

$$\smash[t]{f(x) = (3x^2+2x-5)\overbrace{(x^2+4x+7)}^{\large 0} + 25 = 25}$$

Remark $ $ If your know modular arithemtic then it is simpler to compute $f(x)\bmod x^2+4x+7,\,$ i.e. use $\,x^2 \equiv -4x-7\,$ to reduce all $x^n,\, n> 2$

This is a generalization of the Remainder Theorem $\ f(a) = f(x)\bmod (x-a)$

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  • $\begingroup$ In $2x^2+2x-5$, it should be $3x^2$. $\endgroup$ Dec 18, 2018 at 16:08
  • $\begingroup$ @SameerBaheti Typo fixed, thanks. $\endgroup$ Dec 18, 2018 at 16:09

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