Use Synthetic Division to determine each quotient (and remainder - if necessary). If possible, use the quotients to write the dividend in factored form (no imaginary values) and list all the zeros. If NOT po with given divisor, write DNE for all applicable answers. Complete work must be shown for credit! \( \frac{x^{4}-3 x^{3}+6 x^{2}-12 x+8}{x-1} \) Quotient: \( x^{3}-2 x^{2}+4 x-8 \) Factored Form: \( (x-1)(x-2)\left(x^{2}+4\right) \) Zeros: \( x= \)

Let's dive right into this! Using synthetic division, you can easily divide polynomials, and here's how it works for your polynomial \(x^{4}-3x^{3}+6x^{2}-12x+8\) with the divisor \(x-1\). 1. We set up synthetic division using the root of the divisor, which is 1. 2. Write down the coefficients: 1, -3, 6, -12, 8. 3. Bring down the leading coefficient: 1. 4. Multiply it by the root (1) and add to the next coefficient: \( -3 + 1 = -2 \). 5. Repeat this process: \( -2 \times 1 = -2 \); \( 6 - 2 = 4 \); \( 4 \times 1 = 4 \); \( -12 + 4 = -8 \); \( -8 \times 1 = -8 \); \( 8 - 8 = 0 \). 6. The final coefficients from synthetic division give you the quotient \(x^{3} - 2x^{2} + 4x - 8\) and a remainder of 0. Now, you can express the dividend in factored form as: \[ (x-1)(x^{3} - 2x^{2} + 4x - 8) \] You can factor \(x^{3} - 2x^{2} + 4x - 8\) further. By grouping, you can identify a factor of \((x-2)\) with remainder \(x^{2}+4\). Thus, the factored form is: \[ (x-1)(x-2)(x^{2}+4) \] The zeros can be found from the factors: - From \(x-1 = 0\), \(x = 1\) - From \(x-2 = 0\), \(x = 2\) - From \(x^{2}+4 = 0\), \(x = \pm 2i\) (not real, so we ignore for real zeros). So, the real zeros are: \[ x = 1, 2 \] **Final Answers:** Quotient: \( x^{3}-2 x^{2}+4 x-8 \) Factored Form: \( (x-1)(x-2)(x^{2}+4) \) Zeros: \( x = 1, 2 \)