Polynomial question.

Question

Why do we have to substitute $x=0$ to find out the constant term in this example : $$p(x+2)=x^2+5x+7.$$

Why don't we set $x$ equal to $-2$?

Answer 1

Of course this depends on which polynomial you are talking about:
Since you stated that "Why do we have to substitute $x=0$" I understand that you are asked to find the constant term of $P(x+2)$

To find out the constant term, which is $7$, in $P(x+2)=x^2+5x+7$ ; you need to get rid of all of the $x$'s that appears in the polynomial..
So, you need to plug $0$ for $x$ in the equation regardless of what is inside of $P(\cdots)$
Thus, when you put $0 $ for $x$ in the polynomial you get $P(0+2)=0^2+(5\times 0)+7=7$

But if you were asked to find the constant term of $P(x)$ you would need to get $P(0)$. So this time you need to plug $-2$ for $x$ in $P(x+2)=x^2+5x+7$ to get $P(0)$

Don't condition yourself as if you always have to find out $P(0)$.. This is a common mistake students do when they start to study polynomials for the first time.

Answer 2

$p(x+2)=x^2+5x+7 = ax^2 + bx + c$

Clearly, for $p(x+2)$ c is 7, which is x independent term and can be obtained by putting x =0.

Answer 3

The independent term of $q(x):=p(x+2)=x^2+5x+7$ is $q(0)=p(2)=0^2+5\cdot0+7=7$.

The independent term of $p(x)$ is $p(0)=p(-2+2)=(-2)^2+5(-2)+7=1$.