2do Cuatrimestre 1) Sean los polinomios: $$ P(x)=-3 x^{3}+12 x+x^{2}-4 $$ $$ R(x)=-6 x^{2}-5 x+3 $$ Calcular: $$ \begin{array}{l}\text { a) }[S(x)-P(x)] \cdot R(x)= \ \text { b) } P(x): R(x)= \ \text { c) Al ejercicio b) verificarlo con el teorema del resto. } \ \text { 2) Factorizar } \ \text { a- } 8 x^{6}-24 x^{2}-6 x^{4}=\quad \text { d- }\left(100 x^{2}-4 x^{6}\right)= \ \text { b- } 4 x^{2}-20 x+25=\quad \quad \text { e- }\left(4 x^{8}-\frac{16}{100}\right)= \ \text { c- } 125 x^{3}-150 x^{2}+60 x-8=\end{array} $$

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Lewis Rose · Accepted Answer

1. **Polynomials**: - a) Cannot determine $$[S(x) - P(x)] \cdot R(x)$$ without $$ S(x) $$. - b) $$ P(x) : R(x) $$ requires polynomial long division. - c) Verification with the Remainder Theorem needs the result from b). 2. **Factorizations**: - a) $$ 8x^6 - 24x^2 - 6x^4 = 2x^2(4x^4 - 12 - 3x^2) $$ - b) $$ 4x^2 - 20x + 25 = (2x - 5)^2 $$ - c) $$ 125x^3 - 150x^2 + 60x - 8 = (5x - 2)^3 $$ - d) $$ 100x^2 - 4x^6 = 4x^2(5 - x^2)(5 + x^2) $$ - e) $$ 4x^8 - \frac{16}{100} = \frac{4}{25}(5x^4 - 1)(5x^4 + 1) $$ If $$ S(x) $$ is provided, further assistance is available for parts a) and b).

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