Answer :

To simplify the expression [tex]N = \frac{132x^7 + 60x^5 - 84x^3}{-12x^3}[/tex], we will divide each term in the numerator by [tex]-12x^3[/tex].

Step-by-step:

  1. Simplify each term:

    • The first term is [tex]\frac{132x^7}{-12x^3}[/tex]. Divide the coefficients: [tex]\frac{132}{-12} = -11[/tex]. Subtract the exponents of [tex]x[/tex]: [tex]x^{7-3} = x^4[/tex]. So, the first term simplifies to [tex]-11x^4[/tex].

    • The second term is [tex]\frac{60x^5}{-12x^3}[/tex]. Divide the coefficients: [tex]\frac{60}{-12} = -5[/tex]. Subtract the exponents of [tex]x[/tex]: [tex]x^{5-3} = x^2[/tex]. So, the second term simplifies to [tex]-5x^2[/tex].

    • The third term is [tex]\frac{-84x^3}{-12x^3}[/tex]. Divide the coefficients: [tex]\frac{-84}{-12} = 7[/tex]. Since the exponents of [tex]x[/tex] are equal, they cancel out, resulting in [tex]7[/tex].

  2. Combine the simplified terms:

    The expression becomes: [tex]-11x^4 - 5x^2 + 7[/tex].

  3. Final answer:

    Therefore, the simplified expression is [tex]-11x^4 - 5x^2 + 7[/tex].

This solution involves simplifying a rational expression by applying basic algebraic operations such as dividing coefficients and subtracting exponents.

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