Students will practice proving polynomial identities using elementary algebra. Here are some example problems. After that, students will learn the following: If two linear equations are equal for all \(x,\) then they must be the same line, and thus, they must have the same slope and \(y\)-intercept. Consequently, if \(ax + b = cx + d,\) for all \(x,\) and \(a\) through \(d\) do not reference \(x,\) then \(a = c\) and \(b = d.\) After understanding this, students will use this knowledge to find the values of variables in identities. For example, given$$5ax + 8 + 3(x - d) = 18x + 14$$
which is true for all \(x,\) what are the values of \(a\) and \(d?\) Here's the solution. Next, students will learn this generalizes to polynomials. For example, if \(ax^2 + bx + c = dx^2 + ex + f,\) for all \(x,\) and \(a\) through \(f\) do not reference \(x,\) then \(a = d,\) \(b = e,\) and \(c = f.\) Suppose we are given$$2x^2 - 20x + c = a(x - b)^2 + 3b$$
which is true for all \(x.\) Then what's the value of \(c?\) Here's the solution.