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Sagot :
Claro, podemos proceder con la suma de los dos polinomios. Primero, vamos a escribir los polinomios por separado:
[tex]\[ P_1(x) = \frac{3}{4} a^{\frac{1}{2}} x^2 - \frac{2}{3} x^{\frac{1}{3}} \][/tex]
[tex]\[ P_2(x) = \frac{1}{4} a^{\frac{1}{2}} x^2 + \frac{8}{3} x^{\frac{1}{3}} \][/tex]
El objetivo es sumar [tex]\( P_1(x) \)[/tex] y [tex]\( P_2(x) \)[/tex].
1. Identificar Términos Semejantes:
- Términos en [tex]\(x^2\)[/tex]:
- De [tex]\(P_1(x)\)[/tex]: [tex]\(\frac{3}{4} a^{\frac{1}{2}} x^2\)[/tex]
- De [tex]\(P_2(x)\)[/tex]: [tex]\(\frac{1}{4} a^{\frac{1}{2}} x^2\)[/tex]
- Términos en [tex]\(x^{\frac{1}{3}}\)[/tex]:
- De [tex]\(P_1(x)\)[/tex]: [tex]\(-\frac{2}{3} x^{\frac{1}{3}}\)[/tex]
- De [tex]\(P_2(x)\)[/tex]: [tex]\(\frac{8}{3} x^{\frac{1}{3}}\)[/tex]
2. Sumar los términos semejantes:
- Para los términos en [tex]\(x^2\)[/tex]:
[tex]\[ \left( \frac{3}{4} a^{\frac{1}{2}} x^2 \right) + \left( \frac{1}{4} a^{\frac{1}{2}} x^2 \right) \][/tex]
[tex]\[ = \left( \frac{3}{4} + \frac{1}{4} \right) a^{\frac{1}{2}} x^2 \][/tex]
[tex]\[ = \left( \frac{3 + 1}{4} \right) a^{\frac{1}{2}} x^2 \][/tex]
[tex]\[ = \left( \frac{4}{4} \right) a^{\frac{1}{2}} x^2 \][/tex]
[tex]\[ = a^{\frac{1}{2}} x^2 \][/tex]
- Para los términos en [tex]\(x^{\frac{1}{3}}\)[/tex]:
[tex]\[ \left( -\frac{2}{3} x^{\frac{1}{3}} \right) + \left( \frac{8}{3} x^{\frac{1}{3}} \right) \][/tex]
[tex]\[ = \left( -\frac{2}{3} + \frac{8}{3} \right) x^{\frac{1}{3}} \][/tex]
[tex]\[ = \left( \frac{8 - 2}{3} \right) x^{\frac{1}{3}} \][/tex]
[tex]\[ = \left( \frac{6}{3} \right) x^{\frac{1}{3}} \][/tex]
[tex]\[ = 2 x^{\frac{1}{3}} \][/tex]
3. Resultado Final de la Suma:
Sumando ambos resultados obtenidos, tenemos que:
[tex]\[ P_1(x) + P_2(x) = a^{\frac{1}{2}} x^2 + 2 x^{\frac{1}{3}} \][/tex]
Por lo tanto, la suma de los polinomios es:
[tex]\[ \boxed{a^{\frac{1}{2}} x^2 + 2 x^{\frac{1}{3}}} \][/tex]
[tex]\[ P_1(x) = \frac{3}{4} a^{\frac{1}{2}} x^2 - \frac{2}{3} x^{\frac{1}{3}} \][/tex]
[tex]\[ P_2(x) = \frac{1}{4} a^{\frac{1}{2}} x^2 + \frac{8}{3} x^{\frac{1}{3}} \][/tex]
El objetivo es sumar [tex]\( P_1(x) \)[/tex] y [tex]\( P_2(x) \)[/tex].
1. Identificar Términos Semejantes:
- Términos en [tex]\(x^2\)[/tex]:
- De [tex]\(P_1(x)\)[/tex]: [tex]\(\frac{3}{4} a^{\frac{1}{2}} x^2\)[/tex]
- De [tex]\(P_2(x)\)[/tex]: [tex]\(\frac{1}{4} a^{\frac{1}{2}} x^2\)[/tex]
- Términos en [tex]\(x^{\frac{1}{3}}\)[/tex]:
- De [tex]\(P_1(x)\)[/tex]: [tex]\(-\frac{2}{3} x^{\frac{1}{3}}\)[/tex]
- De [tex]\(P_2(x)\)[/tex]: [tex]\(\frac{8}{3} x^{\frac{1}{3}}\)[/tex]
2. Sumar los términos semejantes:
- Para los términos en [tex]\(x^2\)[/tex]:
[tex]\[ \left( \frac{3}{4} a^{\frac{1}{2}} x^2 \right) + \left( \frac{1}{4} a^{\frac{1}{2}} x^2 \right) \][/tex]
[tex]\[ = \left( \frac{3}{4} + \frac{1}{4} \right) a^{\frac{1}{2}} x^2 \][/tex]
[tex]\[ = \left( \frac{3 + 1}{4} \right) a^{\frac{1}{2}} x^2 \][/tex]
[tex]\[ = \left( \frac{4}{4} \right) a^{\frac{1}{2}} x^2 \][/tex]
[tex]\[ = a^{\frac{1}{2}} x^2 \][/tex]
- Para los términos en [tex]\(x^{\frac{1}{3}}\)[/tex]:
[tex]\[ \left( -\frac{2}{3} x^{\frac{1}{3}} \right) + \left( \frac{8}{3} x^{\frac{1}{3}} \right) \][/tex]
[tex]\[ = \left( -\frac{2}{3} + \frac{8}{3} \right) x^{\frac{1}{3}} \][/tex]
[tex]\[ = \left( \frac{8 - 2}{3} \right) x^{\frac{1}{3}} \][/tex]
[tex]\[ = \left( \frac{6}{3} \right) x^{\frac{1}{3}} \][/tex]
[tex]\[ = 2 x^{\frac{1}{3}} \][/tex]
3. Resultado Final de la Suma:
Sumando ambos resultados obtenidos, tenemos que:
[tex]\[ P_1(x) + P_2(x) = a^{\frac{1}{2}} x^2 + 2 x^{\frac{1}{3}} \][/tex]
Por lo tanto, la suma de los polinomios es:
[tex]\[ \boxed{a^{\frac{1}{2}} x^2 + 2 x^{\frac{1}{3}}} \][/tex]
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