Theory of Polynomial Functions Help (Finding Polynomials)

Servatis

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Mar 26, 2014
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Find a polynomial with real coefficients that has the given zeros. -1 and 5-2i.

P(x)= (x+1)[x-(5-2i)][x-(5+2i)]
P(x)= (x+1)[(x-5)^2-(2i)^2]

My 'help me solve this' jumps to...

P(x)= (x+1)[x^2-10x+25+4]

My question is how are they getting that expression within the brackets? I keep getting stuck on that. I have this feeling it's really obvious. :/
 
P(x)= (x+1)[x-(5-2i)][x-(5+2i)]
P(x)= (x+1)[(x-5)^2-(2i)^2]

My 'help me solve this' jumps to...

P(x)= (x+1)[x^2-10x+25+4]

My question is how are they getting that expression within the brackets?

Hi Servatis:

If "that expression" refers to the blue one, then they squared x-5 to get the first three terms, and they also subtracted the square of 2i to get the fourth term.

(x-5)*(x-5) - (2i)*(2i)

(x^2 - 10x + 25) - (-4)

:)

PS: The red expression comes from recognizing the special factoring pattern called 'Difference of Squares'.

One could also expand the initial expression (in your first line above) to arrive at x^2+10x+29 directly, by applying the FOIL algorithm twice.
 
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