Factorise \( 9 x^{3}-27 x^{2}+x-3 \)

To factorize the polynomial \( 9x^{3} - 27x^{2} + x - 3 \), we can group the terms: 1. Group: \( (9x^{3} - 27x^{2}) + (x - 3) \) 2. Factor out common factors from each group: \( 9x^{2}(x - 3) + 1(x - 3) \) 3. Now factor out the common binomial \( (x - 3) \): \[ (x - 3)(9x^{2} + 1) \] So the factorized form is \( (x - 3)(9x^{2} + 1) \). Now let's explore this fun polynomial further! Among the notable elements in the factorization, \( 9x^{2} + 1 \) doesn't factor over real numbers since it's always positive, which means it might help understand the shapes and behaviors of polynomial graphs. Since \( x - 3 = 0 \) gives us a real root, this helps us discover where the polynomial touches or crosses the x-axis! Lastly, polynomial expressions like this one can be super handy in physics, especially when determining trajectories of projectiles. Understanding how to manipulate such equations helps in calculating maximum heights, distances traveled, and the effects of gravity, making these mathematical techniques vital for real-world applications!