hard factoring question

Question

Which of the following is NOT true if $p^2 = p + 1$ and p is a real number
A. $p^3 = p^2 + p$
B. $p^4 = p^3 + p + 1$
C. $p^3 = 2p + 1$
D. $ p^3 + p^2 = p + 1$
E. $p = \frac{1 } {p−1}$

I tried factoring it but it did not work.

Answer 1

In D you have $p^3+p^2=p+1$, but $p^2=p+1$ so $$p^3+p^2=p^2$$ $$p^3=0$$ $$p=0$$ but this is not true from $p^2=p+1$

Answer 2

Just manipulate each expression but some cases do follow each other.

As $p^2=p+1$, we have $$p^3=p (p+1) =p^2+p =(p+1) +p =2p+1$$ So $A $ and $C $ are correct.

We have $$p^2-p = p (p-1)=1\Rightarrow p=\frac {1}{p-1} $$, so $E $ follows.

Squaring we get, $$p^4=p^2+2p+1 =(p^2+p)+(p+1) =p^3+p+1$$, so $C $ is correct (from $A $, which is correct, we use $p^3=p^2+p $).

From $C $, we can see clearly that $D $ doesn't follow and is hence the only incorrect option. Hope it helps.