SOLUTION: Hello, Can someone please help me with this? The degree three polynomial f(x) with real coefficients and leading coefficient 1, has 4 and 3 + i among its roots. Express f(x)

Question 46265: Hello,
Can someone please help me with this?
The degree three polynomial f(x) with real coefficients and leading coefficient 1, has 4 and 3 + i among its roots. Express f(x) as a product of linear and quadratic polynomials with real coefficients.
I have come up with two different answers and I am really confused:
Is one of these answers correct?
f(x)=(x+4)(x^2+6x+10) or f(x) = (x-4)(x^2-6x+10)
Thank you
V/r
Doug
Answer by venugopalramana(3286) (Show Source): You can put this solution on YOUR website!
Can someone please help me with this?
The degree three polynomial f(x) with real coefficients and leading coefficient 1, has 4 and 3 + i among its roots. Express f(x) as a product of linear and quadratic polynomials with real coefficients.
since coefficients are real if 3+i is root ,then its conjugate 3-i is also a root.hence
f(x)=k(x-4)(x-3-i)(x-3+i) =0
since coefficient of x^3 is 1,k=1
f(x)=(x-4){(x-3)^2-i^2}=(x-4)(x^2-6x+9+1)=0
=x^3-6x^2+10x-4x^2+24x-40=0
=x^3-10x^2+34x-40=0
I have come up with two different answers and I am really confused:
Is one of these answers correct?
f(x)=(x+4)(x^2+6x+10)...NO..4 IS ROOT MEANS X=4...OR...X-4=0
SIMILARLY.....3+I IS A ROOT MEANS X=3+I...OR...X-3-I=0
or f(x) = (x-4)(x^2-6x+10)..THIS IS CORRECT
Thank you
V/r
Doug
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