Question

show that x+a is a factor of the polynomial p(x) = (x+a)^4 + (x+c)^2 - (a-c)^2

Answer 1

Show x+a is factor of

$P(x)=(x+a)^4+(x+c)^2-(a-c)^2$

We are going to simplify the expressions (x+c)^2 and (a-c)^2:

$\begin{gathered} (x+c)^2=x^2+2x*c+c^2 \\ (a-c)^2=a^2-2a*c+c^2 \end{gathered}$

Sustituing:

$P(x)=(x+a)^4+x^2+2xc+c^2-(a^2-2ac+c^2)$$P(x)=(x+a)^4+x^2+2xc+c^2-a^2+2ac-c^2$

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:

$P(-a)=(-a+a)^4+(-a)^2+2(-a)c+c^2-a^2+2ac-c^2$$P(-a)=(0)^4+a^2-2ac+c^2-a^2+2ac-c^2$

Simplifying:

$P(-a)=0$

Therefore x+a is a factor of the polynomial P(x).