How do you form a polynomial f(x)with real coefficients having given degree and zeros? Degree 4; zeros -5+2i; 3 multiplicity 2 How do you form a polynomial f(x)with real coefficients having given degree and zeros? Degree 5; zeros:-4; -i; -3+i

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Answer 1 · 2018-07-17T18:41:20

1)#f(x)=x^4+4x^3-22x^2-84x+261#
2)#f(x)=x^5+10x^4+ 3 x^3+50x^2+34x+40#

1)#x=-5+2i, x=-5-2i, x=3, x=3#

#f(x)=(x+5-2i)(x+5+2i)(x-3)^2#

Let:

#(x+5-2i)(x+5+2i)=A, (x-3)^2=B#

#A=x^2+5x+2ix+5x+25+10i-2ix-10i-4i^2#

But: #=>i^2=-1#

#:.# #A=x^2+10x+29#

#B=x^2-6x+9#

#f(x)=A*B=(x^2+10x+29)(x^2-6x+9)#

#f(x)=x^4-6x^3+9x^2+10x^3-60x^2+90x+29x^2-174x+261#

#f(x)=x^4+4x^3-22x^2-84x+261#

2)#x=-4, x=+-(i), x=(-3+-i)#

#f(x)=(x+4)(x-i)(x+i)(x+3-i)(x+3+i)#

#f(x)=(x+4)(x^2+1)(x^2+6x+10)#

#f(x)=x^5+10x^4+ 3 x^3+50x^2+34x+40#