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8.4: Ejercicios

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    Ejercicio\(\PageIndex{1}\)

    Dividir por división larga.

    1. \(\dfrac{x^3-4x^2+2x+1}{x-2}\)
    2. \(\dfrac{x^3+6x^2+7x-2}{x+3}\)
    3. \(\dfrac{x^2+7x-4}{x+1}\)
    4. \(\dfrac{x^3+3x^2+2x+5}{x+2}\)
    5. \(\dfrac{2x^3+x^2+3x+5}{x-1}\)
    6. \(\dfrac{2x^4+7x^3+x+3}{x+5}\)
    7. \(\dfrac{2x^4-31x^2-13}{x-4}\)
    8. \(\dfrac{x^3+27}{x+3}\)
    9. \(\dfrac{3x^4+7x^3+5x^2+7x+4}{3x+1}\)
    10. \(\dfrac{8x^3+18x^2+21x+18}{2x+3}\)
    11. \(\dfrac{x^3+3x^2-4x-5}{x^2+2x+1}\)
    12. \(\dfrac{x^5+3x^4-20}{x^2+3}\)
    Contestar
    1. \(x^{2}-2 x-2-\dfrac{3}{x-2}\)
    2. \(x^{2}+3 x-2+\dfrac{4}{x+3}\)
    3. \(x+6-\dfrac{10}{x+1}\)
    4. \(x^{2}+x+\dfrac{5}{x+2}\)
    5. \(2 x^{2}+3 x+6+\dfrac{11}{x-1}\)
    6. \(2 x^{3}-3 x^{2}+15 x-74+\dfrac{373}{x+5}\)
    7. \(2 x^{3}+8 x^{2}+x+4+\dfrac{3}{x-4}\)
    8. \(x^{2}-3 x+9\)
    9. \(x^{3}+2 x^{2}+x+2+\dfrac{2}{3 x+1}\)
    10. \(4 x^{2}+3 x+6\)
    11. \(x+1-\dfrac{7 x+6}{x^{2}+2 x+1}\)
    12. \(x^{3}+3 x^{2}-3 x-9+\dfrac{9 x+7}{x^{2}+3}\)

    Ejercicio\(\PageIndex{2}\)

    Encuentra el resto al dividir\(f(x)\) por\(g(x)\).

    1. \(f(x)=x^3+2x^2+x-3, \quad g(x)=x-2\)
    2. \(f(x)=x^3-5x+8, \quad g(x)=x-3\)
    3. \(f(x)=x^5-1, \quad g(x)=x+1\)
    4. \(f(x)=x^5+5x^2-7x+10, \quad g(x)=x+2\)
    Contestar
    1. resto\(r = 15\)
    2. \(r = 20\)
    3. \(r = -2\)
    4. \(r = 12\)

    Ejercicio\(\PageIndex{3}\)

    Determinar si lo dado\(g(x)\) es un factor de\(f(x)\). Si es así, nombra la raíz correspondiente de\(f(x)\).

    1. \(f(x)=x^2+5x+6, \quad g(x)=x+3\)
    2. \(f(x)=x^3-x^2-3x+8, \quad g(x)=x-4\)
    3. \(f(x)=x^4+7x^3+3x^2+29x+56, \quad g(x)=x+7\)
    4. \(f(x)=x^{999}+1, \quad g(x)=x+1\)
    Contestar
    1. sí,\(g(x)\) es un factor de\(f(x)\), la raíz de\(f(x)\) es\(x = −3\)
    2. \(g(x)\)no es un factor de\(f(x)\)
    3. \(g(x)\)es un factor de\(f(x)\), la raíz de\(f(x)\) es\(x = −7\)
    4. \(g(x)\)es un factor de\(f(x)\), la raíz de\(f(x)\) es\(x = −1\)

    Ejercicio\(\PageIndex{4}\)

    Comprobar que los números dados para\(x\) son raíces de\(f(x)\) (ver Observación). Si los números\(x\) son de hecho raíces, entonces usa esta información para factorial tanto\(f(x)\) como sea posible.

    1. \(f(x)=x^3-2x^2-x+2, \quad x=1\)
    2. \(f(x)=x^3-6x^2+11x-6, \quad x=1, x=2, x=3\)
    3. \(f(x)=x^3-3x^2+x-3, \quad x=3\)
    4. \(f(x)=x^3+6x^2+12x+8, \quad x=-2\)
    5. \(f(x)=x^3+13x^2+50x+56, \quad x=-3, x=-4\)
    6. \(f(x)=x^3+3x^2-16x-48, \quad x=2, x=-4\)
    7. \(f(x)=x^5+5x^4-5x^3-25x^2+4x+20, \quad x=1, x=-1, \quad x=2, x=-2\)
    Contestar
    1. \(f(x)=(x-2)(x-1)(x+1)\)
    2. \(f(x)=(x-1)(x-2)(x-3)\)
    3. \(f(x)=(x-3)(x-i)(x+i)\)
    4. \(f(x)=(x+2)^{3}\)
    5. \(f(x)=(x+2)(x+4)(x+7)\)
    6. \(f(x)=(x-4)(x+3)(x+4)\)
    7. \(f(x)=(x-2)(x-1)(x+1)(x+2)(x+5)\)

    Ejercicio\(\PageIndex{5}\)

    Dividir mediante el uso de división sintética.

    1. \(\dfrac{2x^3+3x^2-5x+7}{x-2}\)
    2. \(\dfrac{4x^3+3x^2-15x+18}{x+3}\)
    3. \(\dfrac{x^3+4x^2-3x+1}{x+2}\)
    4. \(\dfrac{x^4+x^3+1}{x-1}\)
    5. \(\dfrac{x^5+32}{x+2}\)
    6. \(\dfrac{x^3+5x^2-3x-10}{x+5}\)
    Contestar
    1. \(2 x^{2}+7 x+9+\dfrac{25}{x-2}\)
    2. \(4 x^{2}-9 x+12-\dfrac{18}{x+3}\)
    3. \(x^{2}+2 x-7+\dfrac{15}{x+2}\)
    4. \(x^{3}+2 x^{2}+2 x+2+\dfrac{3}{x-1}\)
    5. \(x^{4}-2 x^{3}+4 x^{2}-8 x+16\)
    6. \(x^{2}-3+\dfrac{5}{x+5}\)

    This page titled 8.4: Ejercicios is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Thomas Tradler and Holly Carley (New York City College of Technology at CUNY Academic Works) via source content that was edited to the style and standards of the LibreTexts platform.