What is the factored form of the polynomial \( f(x)=27 x^{3}-108 x^{2}+144 x-64 \) ? \( f(x)=(a x-b)(c x-d)(e x-f) \) \( a= \) \( \square \)

To factor the polynomial \( f(x) = 27x^{3} - 108x^{2} + 144x - 64 \) into the form \( f(x) = (a x - b)(c x - d)(e x - f) \), let's follow the steps below: ### Step 1: Identify a Root First, we look for rational roots using the Rational Root Theorem. Possible roots are factors of the constant term divided by factors of the leading coefficient. Testing \( x = \frac{4}{3} \): \[ f\left(\frac{4}{3}\right) = 27\left(\frac{4}{3}\right)^3 - 108\left(\frac{4}{3}\right)^2 + 144\left(\frac{4}{3}\right) - 64 = 0 \] So, \( x = \frac{4}{3} \) is a root. ### Step 2: Factor Out \( (3x - 4) \) Since \( x = \frac{4}{3} \) is a root, \( (3x - 4) \) is a factor. Perform polynomial division or use synthetic division to divide \( f(x) \) by \( (3x - 4) \): \[ f(x) = (3x - 4)(9x^2 - 24x + 16) \] ### Step 3: Factor the Quadratic Next, factor the quadratic \( 9x^2 - 24x + 16 \): \[ 9x^2 - 24x + 16 = (3x - 4)^2 \] ### Step 4: Combine the Factors Putting it all together: \[ f(x) = (3x - 4)(3x - 4)^2 = (3x - 4)^3 \] ### Conclusion The factored form of the polynomial is \( f(x) = (3x - 4)^3 \). Comparing this to the desired form \( (a x - b)(c x - d)(e x - f) \), we see that: \[ a = 3 \] **Answer:** \( a = 3 \)