Skip to main content
Query

9.2: Panua Maneno ya busara

  • Page ID
    164711
  • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

    Ufafanuzi: Kuzidisha Maneno ya busara

    Ili kuzidisha maneno ya busara, kuzidisha maneno ya nambari na kuzidisha maneno ya denominator. Kisha, ikiwa inawezekana, kurahisisha kwa kuzingatia namba zote na denominator na kuondoa mambo ya kawaida.

    Jaribu kutumia sababu kwa kikundi wakati unapofanya kazi na polynomial ya\(4\) muda mrefu.

    Kumbuka: Madhehebu ya kawaida hayahitajiki wakati wa kuzidisha maneno ya busara!

    Panua maneno ya busara:

    1. \(\dfrac{10t}{6t − 12} ∗ \dfrac{20t − 40}{30t^3}\)
    2. \(\dfrac{7x + 14}{ 2x^2 − 8} ∗ \dfrac{x^2 + 3x − 10}{14x + 21}\)
    3. \(\dfrac{3x^3 − 24}{2x^2 − 14x + 20} ∗ \dfrac{4x^3 − 20x^2 + 3x − 15}{3x^2 + 6x + 12}\)

    Suluhisho

    1. \(\begin{array} &&\dfrac{10t}{6t − 12} ∗ \dfrac{20t − 40}{30t^3} &\text{Example problem} \\ &\dfrac{10t}{6t − 12} ∗ \dfrac{20(t − 2)}{30t^3} &\text{Factor all numerators and denominators} \\ &\dfrac{(2)(5)(t)(2)(2)(5)(t − 2)}{(2)(3)(t − 2)(2)(3)(5)(t)(t^2)} &\text{Factor numbers into prime factors and write with one division bar} \\ &\dfrac{\cancel{(2)}(5)\cancel{(t)}(2)(2)\cancel{(5)}\cancel{(t − 2)}}{\cancel{(2)}(3)\cancel{(t − 2)}(2)(3)\cancel{(5)}\cancel{(t)}(t^2)} &\text{Remove common factors} \\ &\dfrac{(5)(2)(2)}{(3)(2)(3)(t^2)} &\text{Simplify} \\ &\dfrac{20}{18t^2} &\text{Final answer (it is good practice to show final answer as factored as possible)} \end{array} \)
    1. \(\begin{array} &&\dfrac{7x + 14}{2x^2 − 8} ∗ \dfrac{x^2 + 3x − 10}{14x + 21} &\text{Example problem} \\ &\dfrac{7(x + 2)}{2(x^2 − 4)} ∗ \dfrac{(x + 5)(x − 2)}{7(2x + 3)} &\text{Factor all numerators and denominators} \\ &\dfrac{7(x + 2)(x + 5)(x − 2)}{2(x + 2)(x − 2)(7)(2x + 3)} &\text{Further factor algebraic expressions and numbers into prime factors and write with one division bar} \\ &\dfrac{\cancel{7}\cancel{(x + 2)}(x + 5)(x − 2)}{2\cancel{(x + 2)}(x − 2)\cancel{(7)}(2x + 3)} &\text{Remove common factors} \\ &\dfrac{(x + 5)(x − 2)}{2(x − 2)(2x + 3)} &\text{Final answer} \end{array}\)
    1. \(\begin{array} &&\dfrac{3x^3 − 24}{2x^2 − 14x + 20} ∗ \dfrac{4x^3 − 20x^2 + 3x − 15}{3x^2 + 6x + 12} &\text{Example problem} \\ &\dfrac{3(x^3 − 8)}{2(x^2 − 7x + 10)} ∗ \dfrac{4x^2 (x − 5) + 3(x − 5)}{3(x^2 + 2x + 4)} &\text{Factor all numerators and denominators. Use factor by grouping for the \(4\)-mrefu polynomial.}\\ &\ dfrac {3 (x - 2) (x ^ 2 + 2x + 4)} {2 (x - 5) (x - 2)}\ dfrac {(4x^2 + 3) (x - 5)} {3 (x + 2) (x + 2)} &\ maandishi {Zaidi ya hayo maneno ya aljebraic}\\ &\ frac {3 (x - 2) (x ^ 2 + 4) (4x^ 2 + 3) (x - 5)} {2 (x - 5) (x - 5) (x - 2) (3) (x + 2) (x + 2)} &\ maandishi {Sababu zaidi ya aljebraic maneno na namba katika mambo kuu na kuandika kwa bar moja ya mgawanyiko}\\ &\ dfrac {\ kufuta {3}\ kufuta {(x - 2)} (x ^ 2 + 2x 4) (4x^ 2 + 3)\ kufuta {(x - 5)} {2\ kufuta {(x + 2)}\ kufuta {(x 4" 2)}\ kufuta {(3)} (x + 2)} (x + 2)} &\ maandishi {Ondoa mambo ya kawaida}\\ &\ dfrac {(x ^ 2 + 2x + 4) (4x^ 2 + 3)} {2 (x + 2) (x + 2)} & amp;\ maandishi {jibu la mwisho}\\ &\ dfrac {(x ^ 2 + 2x + 4) (4x^ 2 + 3)} {2 (x + 2) ^2} &\ maandishi {Jibu la mwisho pia inaweza kuandikwa katika fomu hii}\ mwisho {safu}\)

    Panua maneno ya busara:

    1. \(\dfrac{x^2 + 4x + 3}{2x^2 − x − 10} ∗ \dfrac{2x^2 + 4x^3}{x^2 + 3x} ∗ \dfrac{x}{x^2 + 3x + 2}\)
    2. \(\dfrac{x^2 + 2x − 15}{x^2 − 4x − 45} ∗ \dfrac{x^2 − 5x − 36}{x^2 + x − 12}\)
    3. \(\dfrac{x^2 + 3x − 40}{x^2 + 2x − 35} ∗ \dfrac{x^2 + 3x − 18}{4x^2 − 5x − 32x + 40}\)