4. Aplicar la división de polinomios en los siguientes ejercicios: \( \begin{array}{l}\text { A. }\left(3 x^{3}-5 x^{2}+x-2\right) \div(x+2) \\ \text { 5. Simplificar las expresiones indicadas y expresar los resultados con exponentes positivos. } \\ \begin{array}{l}\text { A. }\left(x^{3}\left(-2 x^{2}+4 x-3 x^{2}-2\right) \div(2 x+1)\right. \\ \text { B. }-3(x-1)(x+1)^{-4} \\ \text { C. }(x+1)^{-2} \\ \text { D. } \frac{a^{-4} b^{2} c^{-1}}{a^{0} b^{2} c^{2}} \\ \text { E. }\left(\frac{z^{-4} a^{-1} b^{2}}{4^{-1} a^{-1} b^{-1}}\right)^{-2}\end{array}\end{array} \) 6. Resolver

Divide the polynomials by following steps: - step0: Using Long Division to Divide Polynomials: \(\left(3x^{3}-5x^{2}+x-2\right)\div \left(x+2\right)\) - step1: Calculate: \(\left(3x^{2}-11x+23\right)\text{ }\textrm{remainder}\text{ }\left(-48\right)\) Calculate or simplify the expression \( (z^{-4} a^{-1} b^{2} / (4^{-1} a^{-1} b^{-1}))^{-2} \). Simplify the expression by following steps: - step0: Solution: \(\left(\frac{z^{-4}a^{-1}b^{2}}{\left(4^{-1}a^{-1}b^{-1}\right)}\right)^{-2}\) - step1: Remove the parentheses: \(\left(\frac{z^{-4}a^{-1}b^{2}}{4^{-1}a^{-1}b^{-1}}\right)^{-2}\) - step2: Multiply by \(a^{-n}:\) \(\left(\frac{z^{-4}a^{-1}b^{2}ab}{4^{-1}}\right)^{-2}\) - step3: Multiply: \(\left(\frac{z^{-4}b^{3}}{4^{-1}}\right)^{-2}\) - step4: Divide the terms: \(\left(\frac{4b^{3}}{z^{4}}\right)^{-2}\) - step5: Transform the expression: \(\left(\frac{z^{4}}{4b^{3}}\right)^{2}\) - step6: Rewrite the expression: \(\frac{\left(z^{4}\right)^{2}}{\left(4b^{3}\right)^{2}}\) - step7: Evaluate the power: \(\frac{z^{8}}{16b^{6}}\) Calculate or simplify the expression \( -3(x-1)(x+1)^{-4} \). Simplify the expression by following steps: - step0: Solution: \(-3\left(x-1\right)\left(x+1\right)^{-4}\) - step1: Multiply the terms: \(-3\left(x+1\right)^{-4}\left(x-1\right)\) - step2: Express with a positive exponent: \(-3\times \frac{1}{\left(x+1\right)^{4}}\times \left(x-1\right)\) - step3: Multiply the terms: \(-\frac{3}{\left(x+1\right)^{4}}\times \left(x-1\right)\) - step4: Multiply the terms: \(-\frac{3\left(x-1\right)}{\left(x+1\right)^{4}}\) - step5: Simplify: \(-\frac{3x-3}{\left(x+1\right)^{4}}\) - step6: Evaluate the power: \(-\frac{3x-3}{x^{4}+4x^{3}+6x^{2}+4x+1}\) Calculate or simplify the expression \( (x+1)^{-2} \). Simplify the expression by following steps: - step0: Solution: \(\left(x+1\right)^{-2}\) - step1: Express with a positive exponent: \(\frac{1}{\left(x+1\right)^{2}}\) - step2: Calculate: \(\frac{1}{x^{2}+2x+1}\) Calculate or simplify the expression \( (a^{-4} b^{2} c^{-1}) / (a^{0} b^{2} c^{2}) \). Simplify the expression by following steps: - step0: Solution: \(\frac{\left(a^{-4}b^{2}c^{-1}\right)}{\left(a^{0}b^{2}c^{2}\right)}\) - step1: Remove the parentheses: \(\frac{a^{-4}b^{2}c^{-1}}{a^{0}b^{2}c^{2}}\) - step2: Multiply by \(a^{-n}:\) \(a^{-4}b^{2}c^{-1}a^{0}b^{-2}c^{-2}\) - step3: Evaluate the power: \(a^{-4}b^{2}c^{-1}\times 1\times b^{-2}c^{-2}\) - step4: Rewrite the expression: \(a^{-4}b^{2}c^{-1}b^{-2}c^{-2}\) - step5: Multiply the terms: \(a^{-4}b^{2-2}c^{-1}\times c^{-2}\) - step6: Subtract the numbers: \(a^{-4}b^{0}c^{-1}\times c^{-2}\) - step7: Evaluate the power: \(a^{-4}c^{-1}\times c^{-2}\) - step8: Multiply the terms: \(a^{-4}c^{-1-2}\) - step9: Subtract the numbers: \(a^{-4}c^{-3}\) - step10: Simplify: \(\frac{1}{a^{4}c^{3}}\) Calculate or simplify the expression \( x^{3}(-2 x^{2}+4 x-3 x^{2}-2) / (2 x+1) \). Simplify the expression by following steps: - step0: Solution: \(\frac{x^{3}\left(-2x^{2}+4x-3x^{2}-2\right)}{\left(2x+1\right)}\) - step1: Remove the parentheses: \(\frac{x^{3}\left(-2x^{2}+4x-3x^{2}-2\right)}{2x+1}\) - step2: Subtract the terms: \(\frac{x^{3}\left(-5x^{2}+4x-2\right)}{2x+1}\) - step3: Multiply the terms: \(\frac{-5x^{5}+4x^{4}-2x^{3}}{2x+1}\) Aquí están las soluciones a los ejercicios planteados: ### 4. División de polinomios **A.** Para el polinomio \( (3x^{3}-5x^{2}+x-2) \div (x+2) \): El resultado de la división es: \[ 3x^{2} - 11x + 23 \quad \text{con un residuo de} \quad -48 \] ### 5. Simplificación de expresiones **A.** Simplificando la expresión \( \frac{x^{3}(-2x^{2}+4x-3x^{2}-2)}{2x+1} \): \[ \frac{-5x^{5}+4x^{4}-2x^{3}}{2x+1} \] **B.** Simplificando \( -3(x-1)(x+1)^{-4} \): \[ -\frac{3(x-1)}{x^{4}+4x^{3}+6x^{2}+4x+1} \] **C.** Simplificando \( (x+1)^{-2} \): \[ \frac{1}{x^{2}+2x+1} \] **D.** Simplificando \( \frac{a^{-4}b^{2}c^{-1}}{a^{0}b^{2}c^{2}} \): \[ \frac{1}{a^{4}c^{3}} \] **E.** Simplificando \( \left(\frac{z^{-4}a^{-1}b^{2}}{4^{-1}a^{-1}b^{-1}}\right)^{-2} \): \[ \frac{z^{8}}{16b^{6}} \] Si necesitas más ayuda o aclaraciones sobre algún paso, no dudes en preguntar.