Divide using synthetic division. \[ \left(4 x^{3}-2 x^{2}+5 x-2\right)+(x-1) \] \( \left(4 x^{3}-2 x^{2}+5 x-2\right)+(x-1)=\square \) (Simplify your answer. Do not factor.)
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To solve this problem, we first need to simplify the expression \( (4 x^{3}-2 x^{2}+5 x-2) + (x-1) \). Combine like terms: \[ 4x^3 - 2x^2 + 5x - 2 + x - 1 = 4x^3 - 2x^2 + 6x - 3. \] Next, we can organize our synthetic division using the polynomial \( 4x^3 - 2x^2 + 6x - 3 \). We are dividing this polynomial by \( (x - 1) \), so we use 1 as our synthetic divisor: 1. Write the coefficients: 4, -2, 6, -3. 2. Bring down the 4. 3. Multiply 4 by 1 (the divisor) and add to -2, giving 2. 4. Multiply 2 by 1 and add to 6, giving 8. 5. Multiply 8 by 1 and add to -3, giving 5. The synthetic division gives us a quotient of \( 4x^2 + 2x + 8 \) with a remainder of 5. Therefore, we write the answer in the correct format: \[ (4 x^{3}-2 x^{2}+5 x-2) + (x-1) = 4x^2 + 2x + 8 + \frac{5}{(x-1)}. \]
