Equality of Polynomials
What are equivalent polynomials?
When we have two functions, f(x) and g(x), whereby f(x) = g(x) for all real values for x, we can say that f(x) and g(x) are equivalent functions.
Likewise, in polynomials, when we have two polynomials, P(x) and Q(x), where by P(x) = Q(x) for all real values of x, we can say that P(x) and Q(x) are equivalent polynomials.
For example, let P(x) = (x+1)2 and Q(x) = x2 + 2x + 1. When we substitute any real value of x into both P(x) and Q(x), their values are always the same.
| x | -2 | -1 | 0 | 1 | 2 | 3 |
| P(x) | 1 | 0 | 1 | 4 | 9 | 16 |
| Q(x) | 1 | 0 | 1 | 4 | 9 | 16 |
Are the Polynomials equivalent?
Finding coefficients in equivalent polynomials
When we have equivalent polynomials, whereby some of the coefficients are unknown, we can employ different methods to solve for the unknown coefficients. We can do either of the following (or both!)
Method 1: Substituting suitable values of x in both polynomials. Their values still equal anyway.
Method 2: Expand the expression to compare the respective coefficients.