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) $$ \left(45 a^{7} b^{3}-60 a^{6} b^{2}+30 a^{5} b\right) \div 15 a^{4} b^{3} $$

Question

UpStudy AI · Accepted Answer

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.
a) $$ -4x + dx $$ b) $$ 10x^{2} - 6x + 2 $$ c) $$ 10a - 7b - 9c $$ d) $$ 5x^{2} - 8x - 2 $$ e) $$ 5x^{2} - 8x - 2 $$ f) $$ 3x^{3} + 23x - 6 $$ g) $$ 11a + b - 4 $$ h) $$ 5h^{3} + 5h^{2} - 3h + 28 $$ i) $$ 2a $$ j) $$ 15a^{6}b^{2} - 6a^{5}b^{3} + 3a^{4}b^{4} - 21b^{5}a^{3} $$ k) $$ 10x^{2} - x - 3 $$ l) $$ \frac{3x^{8} + 10x^{7} - 32x^{4} - 27x^{6} + 10x^{3} - 16x^{5} + 2x^{2} + 6x - 1}{x} $$ m) $$ -x^{2} - 2x + 3 $$ n) $$ \frac{3a^{3}b^{2} - 4a^{2}b + 2a}{b^{2}} $$ Si necesitas más ayuda o aclaraciones sobre algún problema, no dudes en preguntar.

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