The statement shown suggests that if a polynomial equation is factorable, then evaluating the function at $f(1)$ or $f(-1)$ should yield a value of 0, which implies that $x = 1$ or $x = -1$ are potential roots or solutions.

This rule applies primarily when checking for simple factors such as $(x - 1)$ or $(x + 1)$. Specifically, it is related to a shortcut for determining whether a polynomial can be factored easily. The rationale behind this is:

• If $f(1) = 0$, then $(x - 1)$ is a factor of the polynomial.
• If $f(-1) = 0$, then $(x + 1)$ is a factor of the polynomial.

This idea can be applied when testing for rational roots of polynomials using the Rational Root Theorem. If the polynomial is factorable, checking these specific points is a good way to begin factoring or to check if $x = 1$ or $x = -1$ are roots.

This rule is helpful in:

1. Polynomial equations where you're testing for linear factors.
2. Situations where you are factoring by trial and error, and you want to test simple potential roots like $1$ or $-1$.
3. Working with polynomials where small integer roots are likely.

Would you like more details or examples on this concept?

Here are five related questions that build on this:

1. What is the Rational Root Theorem, and how does it apply to factoring polynomials?
2. How can synthetic division help verify whether $x = 1$ or $x = -1$ are roots?
3. Are there any conditions where $f(1)$ and $f(-1)$ won't help in factoring a polynomial?
4. How can you factor a polynomial if $f(1)$ and $f(-1)$ are not roots?
5. What other values, aside from 1 and -1, are commonly checked when factoring polynomials?

Tip: When factoring polynomials, always check for small integer roots first as they can simplify the process.