The function f is defined as \( \mathrm{f}(x)=3^{2 x}-28\left(3^{x}\right)+27, x \in \mathbb{R} \). a Write \( \mathrm{f}(x) \) in the form \( \left(3^{x}-a\right)\left(3^{x}-b\right) \) where \( a \) and \( b \) are real constants.

To rewrite the function \( f(x) = 3^{2x} - 28(3^x) + 27 \) in the form \( (3^x - a)(3^x - b) \), we can start by making a substitution. Let \( y = 3^x \). Then, we can rewrite the function as: \[ f(y) = y^2 - 28y + 27 \] Next, we need to factor the quadratic expression \( y^2 - 28y + 27 \). We will look for two numbers that multiply to \( 27 \) (the constant term) and add up to \( -28 \) (the coefficient of \( y \)). The numbers that satisfy these conditions are \( -27 \) and \( -1 \), since: \[ -27 \times -1 = 27 \quad \text{and} \quad -27 + (-1) = -28 \] Thus, we can factor the quadratic as follows: \[ f(y) = (y - 27)(y - 1) \] Now, substituting back \( y = 3^x \), we have: \[ f(3^x) = (3^x - 27)(3^x - 1) \] Therefore, we can express \( f(x) \) in the desired form: \[ f(x) = (3^x - 27)(3^x - 1) \] In conclusion, the constants \( a \) and \( b \) are: \[ a = 27, \quad b = 1 \]