##### Finding a polynomial of a given degree with given zeros

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Find a polynomial f(x) of degree 4 with real coeffiecients and the following zeros.

4, -3, -1+i

Simplify completely.

Oct 16th, 2015

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The zeros of f(x) are

x = 4, x = -3 and x = -1+i

So factors of f(x) are

x - 4,   x + 3,  x - (-1 + i) , x - (-1 - i)

(because if complex roots or zeros always occur in conjugate pairs. So if -1 + i is a root then its complex conjugate -1 - i is also a root of f(x))

Now multiply these four factors to get the polynomial.

$\\ f(x)=(x-4)(x+3)(x-(-1+i))(x-(-1-i))\\ \\ f(x)=(x^2+3x-4x-12)(x+1-i)(x+1+i)\\ \\ f(x)=(x^2-x-12)(x^2+x+ix+x+1+i-ix-i-i^2)\\ \\ f(x)=(x^2-x-12)(x^2+2x+1-(-1))\\ \\ f(x)=(x^2-x-12)(x^2+2x+1+1)\\ \\ f(x)=(x^2-x-12)(x^2+2x+2)\\ \\ f(x)=x^2(x^2+2x+2)-x(x^2+2x+2)-12(x^2+2x+2)\\ \\ f(x)=x^4+2x^3+2x^2-x^3-2x^2-2x-12x^2-24x-24\\ \\ f(x)=x^4+x^3-12x^2-26x-24\\$

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Oct 16th, 2015

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Oct 16th, 2015
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Oct 16th, 2015
Jun 24th, 2017
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