6. Hallar la suma de \( a^{2}+\frac{1}{2} a b,-\frac{1}{4} a b+\frac{1}{2} b^{2},-\frac{1}{4} a b-\frac{1}{5} b^{2} \) \( \begin{array}{l}\text { 7. Restar } a^{4}-3 a^{2}+5 \text { de la suma de } 5 a^{3}+14 a^{2}-19 a+8 \text {, } \\ a^{5}+9 a-1,-a^{4}+3 a^{2}-1\end{array} \) 8. Determinar la siguiente operación de expresiones: a. \( \left(3 x^{4}-6 x^{2}+7 x\right)+\left(-6 x^{4}+3 x^{2}-3 x+1\right)-\left(-5 x^{4}-x^{3}-3 x^{2}-x+1\right)= \) 9. Si la suma de dos polinomios es \( 3 x^{2}-x+7 \) y uno de los polinomios es \( x^{2}+4 \), entonces el otro polinomios es: Eliminar los signos de agrupación y reducir términos 10. semejantes: a. \( 3 b-\{4 a+[-4-(5 b-3 a)+(5 b-a)]\}= \) b. \( 9 a-\{4 b-[2-(2 a-b)-(4 a-6)]\}= \) c. \( -\{-[-x-(x+x y)]-(5 x y-y)+5 x-3 x y\}= \) 11. Despeje de la fórmula la variable especificada

Simplify the expression by following steps: - step0: Simplify: \(-\left(-\left(-x-\left(x+xy\right)\right)-\left(5xy-y\right)+5x-3xy\right)\) - step1: Subtract the terms: \(-\left(-\left(-2x-xy\right)-\left(5xy-y\right)+5x-3xy\right)\) - step2: Multiply the terms: \(-\left(2x+xy-\left(5xy-y\right)+5x-3xy\right)\) - step3: Calculate: \(-\left(7x-7xy+y\right)\) - step4: Multiply the terms: \(-7x+7xy-y\) Expand the expression \( a^{2}+\frac{1}{2} a b -\frac{1}{4} a b+\frac{1}{2} b^{2} -\frac{1}{4} a b-\frac{1}{5} b^{2} \) Simplify the expression by following steps: - step0: Calculate: \(a^{2}+\frac{1}{2}ab-\frac{1}{4}ab+\frac{1}{2}b^{2}-\frac{1}{4}ab-\frac{1}{5}b^{2}\) - step1: Subtract the terms: \(a^{2}+0+\frac{1}{2}b^{2}-\frac{1}{5}b^{2}\) - step2: Remove 0: \(a^{2}+\frac{1}{2}b^{2}-\frac{1}{5}b^{2}\) - step3: Subtract the terms: \(a^{2}+\frac{3}{10}b^{2}\) Expand the expression \( 3 b - (4 a + [-4 - (5 b - 3 a) + (5 b - a)]) \) Simplify the expression by following steps: - step0: Simplify: \(3b-\left(4a+\left(-4-\left(5b-3a\right)+\left(5b-a\right)\right)\right)\) - step1: Remove the parentheses: \(3b-\left(4a+\left(-4-\left(5b-3a\right)+5b-a\right)\right)\) - step2: Calculate: \(3b-\left(4a+\left(-4+2a\right)\right)\) - step3: Remove the parentheses: \(3b-\left(4a-4+2a\right)\) - step4: Add the terms: \(3b-\left(6a-4\right)\) - step5: Remove the parentheses: \(3b-6a+4\) Expand the expression \( (3 x^{4}-6 x^{2}+7 x) + (-6 x^{4}+3 x^{2}-3 x+1) - (-5 x^{4}-x^{3}-3 x^{2}-x+1) \) Simplify the expression by following steps: - step0: Simplify: \(\left(3x^{4}-6x^{2}+7x\right)+\left(-6x^{4}+3x^{2}-3x+1\right)-\left(-5x^{4}-x^{3}-3x^{2}-x+1\right)\) - step1: Remove the parentheses: \(3x^{4}-6x^{2}+7x-6x^{4}+3x^{2}-3x+1-\left(-5x^{4}-x^{3}-3x^{2}-x+1\right)\) - step2: Remove the parentheses: \(3x^{4}-6x^{2}+7x-6x^{4}+3x^{2}-3x+1+5x^{4}+x^{3}+3x^{2}+x-1\) - step3: Calculate: \(2x^{4}+0+7x-3x+1+x^{3}+x-1\) - step4: Remove 0: \(2x^{4}+7x-3x+1+x^{3}+x-1\) - step5: Calculate: \(2x^{4}+5x+x^{3}\) Expand the expression \( (5 a^{3}+14 a^{2}-19 a+8) - (a^{4}-3 a^{2}+5) + (a^{5}+9 a-1) - (-a^{4}+3 a^{2}-1) \) Simplify the expression by following steps: - step0: Simplify: \(\left(5a^{3}+14a^{2}-19a+8\right)-\left(a^{4}-3a^{2}+5\right)+\left(a^{5}+9a-1\right)-\left(-a^{4}+3a^{2}-1\right)\) - step1: Remove the parentheses: \(5a^{3}+14a^{2}-19a+8-\left(a^{4}-3a^{2}+5\right)+a^{5}+9a-1-\left(-a^{4}+3a^{2}-1\right)\) - step2: Remove the parentheses: \(5a^{3}+14a^{2}-19a+8-a^{4}+3a^{2}-5+a^{5}+9a-1-\left(-a^{4}+3a^{2}-1\right)\) - step3: Remove the parentheses: \(5a^{3}+14a^{2}-19a+8-a^{4}+3a^{2}-5+a^{5}+9a-1+a^{4}-3a^{2}+1\) - step4: Calculate: \(5a^{3}+14a^{2}-10a+3+a^{5}\) Expand the expression \( 9 a - (4 b - [2 - (2 a - b) - (4 a - 6)]) \) Simplify the expression by following steps: - step0: Simplify: \(9a-\left(4b-\left(2-\left(2a-b\right)-\left(4a-6\right)\right)\right)\) - step1: Subtract the terms: \(9a-\left(4b-\left(8-6a+b\right)\right)\) - step2: Subtract the terms: \(9a-\left(3b-8+6a\right)\) - step3: Remove the parentheses: \(9a-3b+8-6a\) - step4: Subtract the terms: \(3a-3b+8\) Solve the equation \( 3 x^{2}-x+7 = x^{2}+4 + p \). Solve the equation by following steps: - step0: Solve for \(p\): \(3x^{2}-x+7=x^{2}+4+p\) - step1: Swap the sides: \(x^{2}+4+p=3x^{2}-x+7\) - step2: Move the expression to the right side: \(p=3x^{2}-x+7-\left(x^{2}+4\right)\) - step3: Subtract the terms: \(p=2x^{2}-x+3\) Aquí tienes las soluciones a los problemas planteados: **6. Suma de expresiones:** La suma de las expresiones \( a^{2}+\frac{1}{2} a b \), \(-\frac{1}{4} a b+\frac{1}{2} b^{2} \), y \(-\frac{1}{4} a b-\frac{1}{5} b^{2} \) es: \[ a^{2} + \frac{1}{2} a b - \frac{1}{4} a b + \frac{1}{2} b^{2} - \frac{1}{4} a b - \frac{1}{5} b^{2} = a^{2} + \frac{3}{10} b^{2} \] **7. Resta de polinomios:** Restamos \( a^{4}-3 a^{2}+5 \) de la suma de \( 5 a^{3}+14 a^{2}-19 a+8 \), \( a^{5}+9 a-1 \), y \(-a^{4}+3 a^{2}-1\): \[ (5 a^{3}+14 a^{2}-19 a+8) - (a^{4}-3 a^{2}+5) + (a^{5}+9 a-1) - (-a^{4}+3 a^{2}-1) = 5a^{3}+14a^{2}-10a+3+a^{5} \] **8. Operación de expresiones:** La operación de las expresiones \( (3 x^{4}-6 x^{2}+7 x) + (-6 x^{4}+3 x^{2}-3 x+1) - (-5 x^{4}-x^{3}-3 x^{2}-x+1) \) es: \[ 2x^{4}+5x+x^{3} \] **9. Determinación del polinomio:** Si la suma de dos polinomios es \( 3 x^{2}-x+7 \) y uno de los polinomios es \( x^{2}+4 \), el otro polinomio \( p \) se determina como: \[ p = 3 x^{2} - x + 7 - (x^{2} + 4) = 2x^{2} - x + 3 \] **10. Eliminación de signos de agrupación y reducción de términos:** a. \( 3 b - \{4 a + [-4 - (5 b - 3 a) + (5 b - a)]\} = 3b - 6a + 4 \) b. \( 9 a - \{4 b - [2 - (2 a - b) - (4 a - 6)]\} = 3a - 3b + 8 \) c. \( -\{-[-x - (x + x y)] - (5 x y - y) + 5 x - 3 x y\} = -7x + 7xy - y \) Si necesitas más ayuda o aclaraciones, no dudes en preguntar.