Show x+a is factor of
[tex]P(x)=(x+a)^4+(x+c)^2-(a-c)^2[/tex]We are going to simplify the expressions (x+c)^2 and (a-c)^2:
[tex]\begin{gathered} (x+c)^2=x^2+2x*c+c^2 \\ (a-c)^2=a^2-2a*c+c^2 \end{gathered}[/tex]Sustituing:
[tex]P(x)=(x+a)^4+x^2+2xc+c^2-(a^2-2ac+c^2)[/tex][tex]P(x)=(x+a)^4+x^2+2xc+c^2-a^2+2ac-c^2[/tex]Now we can evaluate if x+a is a factor, x+a is a factor if the polynomial in x+a=0 is a root of p(x), we can see
x=-a
Sustituing:
[tex]P(-a)=(-a+a)^4+(-a)^2+2(-a)c+c^2-a^2+2ac-c^2[/tex][tex]P(-a)=(0)^4+a^2-2ac+c^2-a^2+2ac-c^2[/tex]Simplifying:
[tex]P(-a)=0[/tex]Therefore x+a is a factor of the polynomial P(x).
