Determine whether the expression is a trinomial square. \( r^{2}-9 r+81 \) Is the expression a trinomial square? No Yes

To determine if the expression \( r^{2}-9 r+81 \) is a trinomial square, we should look for its potential factorization. A trinomial square has the form \( (a-b)^2 = a^2 - 2ab + b^2 \). Here, \( a^2 = r^2 \) and \( b^2 = 81 \) implies \( b = 9 \). If we plug these values in, we get \( -2ab = -18r \) instead of \( -9r \), which means it doesn't fit the perfect square format. Therefore, it's not a trinomial square! In practical terms, recognizing trinomial squares aids in simplifying mathematical expressions and solving equations faster. They save time in algebraic manipulations, especially in calculus where optimization or integration of polynomials is involved. So understanding their structure can be a powerful tool for any math enthusiast!