Final answer:
To prove polynomial identities and describe numerical relationships, one can use factoring and mathematical induction with substitution. Factoring simplifies the polynomial, while mathematical induction proves concepts for sequences and series.
Step-by-step explanation:
To prove polynomial identities and use them to describe numerical relationships, there are several methods that can be employed. The options provided suggest three primary techniques: induction, factoring, and mathematical induction and substitution. Among these, the most commonly used techniques are through factoring and using mathematical induction and substitution.
Factoring involves rewriting the polynomial as a product of its factors. This can simplify the problem and reveal hidden patterns or relationships. To prove an identity by factoring, you would factor both sides of the equation and then demonstrate that they are equivalent.
Mathematical induction is a technique where you prove that a statement holds for a base case and then assume it's true for an arbitrary case 'n'. You then prove the case of 'n+1' based on the assumption. This is very effective for proving identities that are dependent on integers, especially sequences and series.
Substitution involves replacing a variable with a known or assumed value to simplify and solve the polynomial equation or to prove an identity. This can be particularly useful when dealing with complex or higher-degree polynomials.