Trabajo de Pensamiento Matemático II a) \( (5 x+d x)+(-2 x-7 x) \) b) \( \left(3 x^{2}-5 x+10\right)+\left(x^{2}+2 x-7\right)+\left(6 x^{2}-3 x-1\right) \) c) \( (2 a+3 b-5 c)+(8 a-10 b-4 c) \) d) \( \left(7 x^{2}-5 x+3\right)-\left(2 x^{2}+3 x+5\right) \) e) \( \left(7 x^{2}-5 x+3\right)-\left(2 x^{2}+3 x+5\right) \) f) \( \left(2 x^{3}-5 x^{2}+3 x-8\right)+\left(5 x^{2}+10 x-3\right)+\left(x^{3}+10 x+5\right) \) g) \( (8 a+b)-(-3 a+4) \) h) \( \left(5 h^{3}+2 h^{2}-3\right)+\left(h^{2}-3 h+11\right)+\left(2 h^{2}+20\right) \) i) \( (a+b)+(a-b) \) j) \( \left(-3 a^{3} b^{2}\right)\left(-5 a^{3}+2 a^{2} b-a b^{2}+7 b^{3}\right) \) K) \( (2 x+1)(5 x-3) \) L) \( \left(3 x^{4}-5 x^{3}-2 x^{2}-6 x+1\right)\left(x^{3}+5 x^{2}-x^{-1}\right) \) m) \( \left(8 x^{5}+16 x^{4}-24 x^{3}\right) \div\left(-8 x^{3}\right) \) n) \( \left(45 a^{7} b^{3}-60 a^{6} b^{2}+30 a^{5} b\right) \div 15 a^{4} b^{3} \)

Simplify the expression by following steps: - step0: Simplify: \(\left(8a+b\right)-\left(-3a+4\right)\) - step1: Remove the parentheses: \(8a+b-\left(-3a+4\right)\) - step2: Remove the parentheses: \(8a+b+3a-4\) - step3: Add the terms: \(11a+b-4\) Expand the expression \( (a+b)+(a-b) \) Simplify the expression by following steps: - step0: Simplify: \(\left(a+b\right)+\left(a-b\right)\) - step1: Remove the parentheses: \(a+b+a-b\) - step2: Add the terms: \(2a\) Expand the expression \( (5 x+d x)+(-2 x-7 x) \) Simplify the expression by following steps: - step0: Simplify: \(\left(5x+dx\right)+\left(-2x-7x\right)\) - step1: Remove the parentheses: \(5x+dx+\left(-2x-7x\right)\) - step2: Subtract the terms: \(5x+dx+\left(-9x\right)\) - step3: Remove the parentheses: \(5x+dx-9x\) - step4: Subtract the terms: \(-4x+dx\) Expand the expression \( \left(8 x^{5}+16 x^{4}-24 x^{3}\right) \div \left(-8 x^{3}\right) \) Simplify the expression by following steps: - step0: Divide the terms: \(\left(8x^{5}+16x^{4}-24x^{3}\right)\div \left(-8x^{3}\right)\) - step1: Rewrite the expression: \(\frac{8x^{5}+16x^{4}-24x^{3}}{-8x^{3}}\) - step2: Rewrite the fraction: \(-\frac{8x^{5}+16x^{4}-24x^{3}}{8x^{3}}\) - step3: Reduce the fraction: \(-\left(x^{2}+2x-3\right)\) - step4: Calculate: \(-x^{2}-2x+3\) Expand the expression \( (2 a+3 b-5 c)+(8 a-10 b-4 c) \) Simplify the expression by following steps: - step0: Simplify: \(\left(2a+3b-5c\right)+\left(8a-10b-4c\right)\) - step1: Remove the parentheses: \(2a+3b-5c+8a-10b-4c\) - step2: Add the terms: \(10a-7b-9c\) Expand the expression \( \left(5 h^{3}+2 h^{2}-3\right)+\left(h^{2}-3 h+11\right)+\left(2 h^{2}+20\right) \) Simplify the expression by following steps: - step0: Simplify: \(\left(5h^{3}+2h^{2}-3\right)+\left(h^{2}-3h+11\right)+\left(2h^{2}+20\right)\) - step1: Remove the parentheses: \(5h^{3}+2h^{2}-3+h^{2}-3h+11+2h^{2}+20\) - step2: Add the terms: \(5h^{3}+5h^{2}+28-3h\) Expand the expression \( (2 x+1)(5 x-3) \) Simplify the expression by following steps: - step0: Multiply the terms: \(\left(2x+1\right)\left(5x-3\right)\) - step1: Apply the distributive property: \(2x\times 5x-2x\times 3+1\times 5x-1\times 3\) - step2: Multiply the terms: \(10x^{2}-6x+5x-3\) - step3: Add the terms: \(10x^{2}-x-3\) Expand the expression \( \left(7 x^{2}-5 x+3\right)-\left(2 x^{2}+3 x+5\right) \) Simplify the expression by following steps: - step0: Simplify: \(\left(7x^{2}-5x+3\right)-\left(2x^{2}+3x+5\right)\) - step1: Remove the parentheses: \(7x^{2}-5x+3-\left(2x^{2}+3x+5\right)\) - step2: Remove the parentheses: \(7x^{2}-5x+3-2x^{2}-3x-5\) - step3: Subtract the terms: \(5x^{2}-8x-2\) Expand the expression \( \left(7 x^{2}-5 x+3\right)-\left(2 x^{2}+3 x+5\right) \) Simplify the expression by following steps: - step0: Simplify: \(\left(7x^{2}-5x+3\right)-\left(2x^{2}+3x+5\right)\) - step1: Remove the parentheses: \(7x^{2}-5x+3-\left(2x^{2}+3x+5\right)\) - step2: Remove the parentheses: \(7x^{2}-5x+3-2x^{2}-3x-5\) - step3: Subtract the terms: \(5x^{2}-8x-2\) Expand the expression \( \left(3 x^{2}-5 x+10\right)+\left(x^{2}+2 x-7\right)+\left(6 x^{2}-3 x-1\right) \) Simplify the expression by following steps: - step0: Simplify: \(\left(3x^{2}-5x+10\right)+\left(x^{2}+2x-7\right)+\left(6x^{2}-3x-1\right)\) - step1: Remove the parentheses: \(3x^{2}-5x+10+x^{2}+2x-7+6x^{2}-3x-1\) - step2: Add the terms: \(10x^{2}-6x+2\) Expand the expression \( \left(45 a^{7} b^{3}-60 a^{6} b^{2}+30 a^{5} b\right) \div 15 a^{4} b^{3} \) Simplify the expression by following steps: - step0: Divide the terms: \(\left(45a^{7}b^{3}-60a^{6}b^{2}+30a^{5}b\right)\div \left(15a^{4}b^{3}\right)\) - step1: Rewrite the expression: \(\frac{45a^{7}b^{3}-60a^{6}b^{2}+30a^{5}b}{15a^{4}b^{3}}\) - step2: Reduce the fraction: \(\frac{3a^{3}b^{2}-4a^{2}b+2a}{b^{2}}\) Expand the expression \( \left(2 x^{3}-5 x^{2}+3 x-8\right)+\left(5 x^{2}+10 x-3\right)+\left(x^{3}+10 x+5\right) \) Simplify the expression by following steps: - step0: Simplify: \(\left(2x^{3}-5x^{2}+3x-8\right)+\left(5x^{2}+10x-3\right)+\left(x^{3}+10x+5\right)\) - step1: Remove the parentheses: \(2x^{3}-5x^{2}+3x-8+5x^{2}+10x-3+x^{3}+10x+5\) - step2: Add the terms: \(3x^{3}+23x-6\) Expand the expression \( \left(-3 a^{3} b^{2}\right)\left(-5 a^{3}+2 a^{2} b-a b^{2}+7 b^{3}\right) \) Simplify the expression by following steps: - step0: Calculate: \(\left(-3a^{3}b^{2}\right)\left(-5a^{3}+2a^{2}b-ab^{2}+7b^{3}\right)\) - step1: Remove the parentheses: \(-3a^{3}b^{2}\left(-5a^{3}+2a^{2}b-ab^{2}+7b^{3}\right)\) - step2: Rewrite the expression: \(-3\left(-5a^{3}+2a^{2}b-ab^{2}+7b^{3}\right)a^{3}b^{2}\) - step3: Multiply the expression: \(-3\left(-5a^{3}+2a^{2}b-ab^{2}+7b^{3}\right)b^{2}a^{3}\) - step4: Rearrange the terms: \(\left(15a^{3}-6a^{2}b+3ab^{2}-21b^{3}\right)b^{2}a^{3}\) - step5: Apply the distributive property: \(15a^{3}b^{2}a^{3}-6a^{2}b\times b^{2}a^{3}+3ab^{2}\times b^{2}a^{3}-21b^{3}\times b^{2}a^{3}\) - step6: Multiply the terms: \(15a^{6}b^{2}-6a^{5}b^{3}+3a^{4}b^{4}-21b^{5}a^{3}\) Expand the expression \( \left(3 x^{4}-5 x^{3}-2 x^{2}-6 x+1\right)\left(x^{3}+5 x^{2}-x^{-1}\right) \) Simplify the expression by following steps: - step0: Multiply the terms: \(\left(3x^{4}-5x^{3}-2x^{2}-6x+1\right)\left(x^{3}+5x^{2}-x^{-1}\right)\) - step1: Rewrite the expression: \(\left(3x^{4}-5x^{3}-2x^{2}-6x+1\right)\times \frac{x^{4}+5x^{3}-1}{x}\) - step2: Multiply the terms: \(\frac{\left(3x^{4}-5x^{3}-2x^{2}-6x+1\right)\left(x^{4}+5x^{3}-1\right)}{x}\) - step3: Multiply the terms: \(\frac{3x^{8}+10x^{7}-32x^{4}-27x^{6}+10x^{3}-16x^{5}+2x^{2}+6x-1}{x}\) Aquí están las soluciones a cada uno de los problemas planteados: a) \( (5 x + d x) + (-2 x - 7 x) \) \[ = -4x + dx \] b) \( \left(3 x^{2}-5 x+10\right) + \left(x^{2}+2 x-7\right) + \left(6 x^{2}-3 x-1\right) \) \[ = 10x^{2} - 6x + 2 \] c) \( (2 a + 3 b - 5 c) + (8 a - 10 b - 4 c) \) \[ = 10a - 7b - 9c \] d) \( \left(7 x^{2}-5 x+3\right) - \left(2 x^{2}+3 x+5\right) \) \[ = 5x^{2} - 8x - 2 \] e) \( \left(7 x^{2}-5 x+3\right) - \left(2 x^{2}+3 x+5\right) \) \[ = 5x^{2} - 8x - 2 \] f) \( \left(2 x^{3}-5 x^{2}+3 x-8\right) + \left(5 x^{2}+10 x-3\right) + \left(x^{3}+10 x+5\right) \) \[ = 3x^{3} + 23x - 6 \] g) \( (8 a + b) - (-3 a + 4) \) \[ = 11a + b - 4 \] h) \( \left(5 h^{3}+2 h^{2}-3\right) + \left(h^{2}-3 h+11\right) + \left(2 h^{2}+20\right) \) \[ = 5h^{3} + 5h^{2} - 3h + 28 \] i) \( (a + b) + (a - b) \) \[ = 2a \] j) \( \left(-3 a^{3} b^{2}\right) \left(-5 a^{3}+2 a^{2} b-a b^{2}+7 b^{3}\right) \) \[ = 15a^{6}b^{2} - 6a^{5}b^{3} + 3a^{4}b^{4} - 21b^{5}a^{3} \] k) \( (2 x + 1)(5 x - 3) \) \[ = 10x^{2} - x - 3 \] l) \( \left(3 x^{4}-5 x^{3}-2 x^{2}-6 x+1\right) \left(x^{3}+5 x^{2}-x^{-1}\right) \) \[ = \frac{3x^{8} + 10x^{7} - 32x^{4} - 27x^{6} + 10x^{3} - 16x^{5} + 2x^{2} + 6x - 1}{x} \] m) \( \left(8 x^{5}+16 x^{4}-24 x^{3}\right) \div \left(-8 x^{3}\right) \) \[ = -x^{2} - 2x + 3 \] n) \( \left(45 a^{7} b^{3}-60 a^{6} b^{2}+30 a^{5} b\right) \div 15 a^{4} b^{3} \) \[ = \frac{3a^{3}b^{2}-4a^{2}b+2a}{b^{2}} \] Si necesitas más ayuda o aclaraciones sobre algún problema, no dudes en preguntar.