Find the value of $3x^4+14x^3+24x^2-6x-10$, when $x=-2+\sqrt{3}i$?

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?

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

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.
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 $$\ f(a) = f(x)\bmod (x-a)$$
• In $2x^2+2x-5$, it should be $3x^2$. Dec 18, 2018 at 16:08