Write the polynomial as a product of linear factors. \( x^{3}-x^{2}+9 x-9 \) \( (x-[?])(x-3 i)(x+[\quad i) \)

To express the polynomial \( x^{3}-x^{2}+9x-9 \) as a product of linear factors, we start by factoring it completely. First, we identify that \( x = 3 \) is a root (you can check this by substituting \( x = 3 \) into the polynomial). Now, we can use polynomial long division or synthetic division to factor \( (x - 3) \) out from \( x^{3}-x^{2}+9x-9 \). After performing the division, we find that the polynomial can be factored as: \[ (x - 3)(x^{2} + 2) \] From this, we notice that \( x^{2} + 2 \) can be factored further into linear factors with complex numbers: \[ (x - 3)(x - i\sqrt{2})(x + i\sqrt{2}) \] Putting it all together, the polynomial written in terms of its linear factors is: \[ (x - 3)(x - i\sqrt{2})(x + i\sqrt{2}) \] This means you will fill the brackets with \( 3 \) and \( \sqrt{2} \) to complete the factorization. Now you have the polynomial factored into linear components splendidly!