factor x^5-x^4+3x-3

To factor the polynomial \( x^5 - x^4 + 3x - 3 \), we can start by checking for rational roots using the Rational Root Theorem. The possible rational roots are factors of the constant term (-3) divided by factors of the leading coefficient (1), which gives us possible roots of \( \pm 1, \pm 3 \). Let's test some of these values: 1. Testing \( x = 1 \): \[ f(1) = 1^5 - 1^4 + 3(1) - 3 = 1 - 1 + 3 - 3 = 0 \] Thus, \( x = 1 \) is a root. Now that we know \( x - 1 \) is a factor, we can perform synthetic division of \( x^5 - x^4 + 3x - 3 \) by \( x - 1 \). ### Synthetic Division: Set up the synthetic division: \[ \begin{array}{r|rrrrrr} 1 & 1 & -1 & 0 & 3 & 0 & -3 \\ & & 1 & 0 & 0 & 3 & 3 \\ \hline & 1 & 0 & 0 & 3 & 3 & 0 \\ \end{array} \] The result from the synthetic division is: \[ x^4 + 0x^3 + 0x^2 + 3x + 3 \text{ or simply } x^4 + 3x + 3. \] So we can write the original polynomial as: \[ x^5 - x^4 + 3x - 3 = (x - 1)(x^4 + 3x + 3). \] ### Factoring \( x^4 + 3x + 3 \): Now we need to check if we can factor \( x^4 + 3x + 3 \) further. To investigate whether \( x^4 + 3x + 3 \) has any rational roots, we can again check possible rational roots, but they tend to be more complex due to the degree. Instead, we could check for irreducibility over the real numbers. #### Checking Factoring: For \( x^4 + 3x + 3 \), we can examine whether it can be expressed as a product of two quadratics or if it has real roots. We calculate the discriminant of \( x^4 + 3x + 3 \): Using numerical estimation, the roots do not yield clean rational roots, suggesting it cannot be factored further over the rationals or reals. ### Conclusion: Thus, the factorization of our original polynomial is: \[ \boxed{(x - 1)(x^4 + 3x + 3)}. \] This is as far as we can factor it without resorting to numerical methods or complex numbers.