Realizar las siguientes operaciones de polinomios, teniendo en cuenta que: $$ \begin{array}{lll}A: x^{3}+2 x^{2} y-5 x y+y^{3} \ \text { B: } 2 x^{3}+3 x y^{2}-2 y^{3} \ C: 4 x^{2}-x y^{2}+y^{3} \ D: x^{3}-x^{2} y+x y^{2}\end{array} $$ $$ \begin{array}{llll}A+B) \times(C-D) & \text { b) }(D \times A)-(C+B) & \text { c) }[(A-C) \times B] \div D & \text { d) }(C-A)+(B \times D)\end{array} $$

Question

UpStudy AI · Accepted Answer

Simplify the expression by following steps:
- step0: Simplify:
  $$\left(\left(4x^{2}-xy^{2}+y^{3}\right)-\left(x^{3}+2x^{2}y-5xy+y^{3}\right)\right)+\left(\left(2x^{3}+3xy^{2}-2y^{3}\right)\left(x^{3}-x^{2}y+xy^{2}\right)\right)$$
- step1: Remove the parentheses:
  $$\left(4x^{2}-xy^{2}+y^{3}-\left(x^{3}+2x^{2}y-5xy+y^{3}\right)\right)+\left(\left(2x^{3}+3xy^{2}-2y^{3}\right)\left(x^{3}-x^{2}y+xy^{2}\right)\right)$$
- step2: Subtract the terms:
  $$\left(4x^{2}-xy^{2}-x^{3}-2x^{2}y+5xy\right)+\left(\left(2x^{3}+3xy^{2}-2y^{3}\right)\left(x^{3}-x^{2}y+xy^{2}\right)\right)$$
- step3: Remove the parentheses:
  $$4x^{2}-xy^{2}-x^{3}-2x^{2}y+5xy+\left(\left(2x^{3}+3xy^{2}-2y^{3}\right)\left(x^{3}-x^{2}y+xy^{2}\right)\right)$$
- step4: Multiply the terms:
  $$4x^{2}-xy^{2}-x^{3}-2x^{2}y+5xy+\left(2x^{3}+3xy^{2}-2y^{3}\right)\left(x^{3}-x^{2}y+xy^{2}\right)$$
- step5: Expand the expression:
  $$4x^{2}-xy^{2}-x^{3}-2x^{2}y+5xy+2x^{6}-2x^{5}y+5x^{4}y^{2}-5x^{3}y^{3}+5x^{2}y^{4}-2y^{5}x$$
Expand the expression $$ ((x^{3}-x^{2} y+x y^{2}) * (x^{3}+2 x^{2} y-5 x y+y^{3})) - (4 x^{2}-x y^{2}+y^{3} + 2 x^{3}+3 x y^{2}-2 y^{3}) $$
Simplify the expression by following steps:
- step0: Simplify:
  $$\left(\left(x^{3}-x^{2}y+xy^{2}\right)\left(x^{3}+2x^{2}y-5xy+y^{3}\right)\right)-\left(4x^{2}-xy^{2}+y^{3}+2x^{3}+3xy^{2}-2y^{3}\right)$$
- step1: Multiply the terms:
  $$\left(x^{3}-x^{2}y+xy^{2}\right)\left(x^{3}+2x^{2}y-5xy+y^{3}\right)-\left(4x^{2}-xy^{2}+y^{3}+2x^{3}+3xy^{2}-2y^{3}\right)$$
- step2: Calculate:
  $$\left(x^{3}-x^{2}y+xy^{2}\right)\left(x^{3}+2x^{2}y-5xy+y^{3}\right)-\left(4x^{2}+2xy^{2}-y^{3}+2x^{3}\right)$$
- step3: Remove the parentheses:
  $$\left(x^{3}-x^{2}y+xy^{2}\right)\left(x^{3}+2x^{2}y-5xy+y^{3}\right)-4x^{2}-2xy^{2}+y^{3}-2x^{3}$$
- step4: Expand the expression:
  $$x^{6}+x^{5}y-5x^{4}y+3x^{3}y^{3}-x^{4}y^{2}+5x^{3}y^{2}-x^{2}y^{4}-5x^{2}y^{3}+xy^{5}-4x^{2}-2xy^{2}+y^{3}-2x^{3}$$
Expand the expression $$ ((x^{3}+2 x^{2} y-5 x y+y^{3} - (4 x^{2}-x y^{2}+y^{3})) * (2 x^{3}+3 x y^{2}-2 y^{3})) / (x^{3}-x^{2} y+x y^{2}) $$
Simplify the expression by following steps:
- step0: Multiply the terms:
  $$\frac{\left(\left(x^{3}+2x^{2}y-5xy+y^{3}-\left(4x^{2}-xy^{2}+y^{3}\right)\right)\left(2x^{3}+3xy^{2}-2y^{3}\right)\right)}{\left(x^{3}-x^{2}y+xy^{2}\right)}$$
- step1: Calculate:
  $$\frac{\left(\left(x^{3}+2x^{2}y-5xy-4x^{2}+xy^{2}\right)\left(2x^{3}+3xy^{2}-2y^{3}\right)\right)}{\left(x^{3}-x^{2}y+xy^{2}\right)}$$
- step2: Multiply the terms:
  $$\frac{\left(x^{3}+2x^{2}y-5xy-4x^{2}+xy^{2}\right)\left(2x^{3}+3xy^{2}-2y^{3}\right)}{\left(x^{3}-x^{2}y+xy^{2}\right)}$$
- step3: Remove the parentheses:
  $$\frac{\left(x^{3}+2x^{2}y-5xy-4x^{2}+xy^{2}\right)\left(2x^{3}+3xy^{2}-2y^{3}\right)}{x^{3}-x^{2}y+xy^{2}}$$
- step4: Factor the expression:
  $$\frac{x\left(x^{2}+2xy-5y-4x+y^{2}\right)\left(2x^{3}+3xy^{2}-2y^{3}\right)}{x\left(x^{2}-xy+y^{2}\right)}$$
- step5: Reduce the fraction:
  $$\frac{\left(x^{2}+2xy-5y-4x+y^{2}\right)\left(2x^{3}+3xy^{2}-2y^{3}\right)}{x^{2}-xy+y^{2}}$$
- step6: Multiply the terms:
  $$\frac{2x^{5}+5x^{3}y^{2}+4x^{2}y^{3}+4x^{4}y-xy^{4}-10yx^{3}-7y^{3}x+10y^{4}-8x^{4}-12x^{2}y^{2}-2y^{5}}{x^{2}-xy+y^{2}}$$
Expand the expression $$ (x^{3}+2 x^{2} y-5 x y+y^{3} + 2 x^{3}+3 x y^{2}-2 y^{3}) * (4 x^{2}-x y^{2}+y^{3} - (x^{3}-x^{2} y+x y^{2})) $$
Simplify the expression by following steps:
- step0: Multiply the terms:
  $$\left(x^{3}+2x^{2}y-5xy+y^{3}+2x^{3}+3xy^{2}-2y^{3}\right)\left(4x^{2}-xy^{2}+y^{3}-\left(x^{3}-x^{2}y+xy^{2}\right)\right)$$
- step1: Calculate:
  $$\left(3x^{3}+2x^{2}y-5xy-y^{3}+3xy^{2}\right)\left(4x^{2}-xy^{2}+y^{3}-\left(x^{3}-x^{2}y+xy^{2}\right)\right)$$
- step2: Calculate:
  $$\left(3x^{3}+2x^{2}y-5xy-y^{3}+3xy^{2}\right)\left(4x^{2}-2xy^{2}+y^{3}-x^{3}+x^{2}y\right)$$
- step3: Apply the distributive property:
  $$3x^{3}\times 4x^{2}-3x^{3}\times 2xy^{2}+3x^{3}y^{3}-3x^{3}\times x^{3}+3x^{3}\times x^{2}y+2x^{2}y\times 4x^{2}-2x^{2}y\times 2xy^{2}+2x^{2}y\times y^{3}-2x^{2}yx^{3}+2x^{2}yx^{2}y-5xy\times 4x^{2}-\left(-5xy\times 2xy^{2}\right)-5xy\times y^{3}-\left(-5xyx^{3}\right)-5xyx^{2}y-y^{3}\times 4x^{2}-\left(-y^{3}\times 2xy^{2}\right)-y^{3}\times y^{3}-\left(-y^{3}x^{3}\right)-y^{3}x^{2}y+3xy^{2}\times 4x^{2}-3xy^{2}\times 2xy^{2}+3xy^{2}\times y^{3}-3xy^{2}x^{3}+3xy^{2}x^{2}y$$
- step4: Multiply the terms:
  $$12x^{5}-6x^{4}y^{2}+3x^{3}y^{3}-3x^{6}+3x^{5}y+8x^{4}y-4x^{3}y^{3}+2x^{2}y^{4}-2x^{5}y+2x^{4}y^{2}-20x^{3}y-\left(-10x^{2}y^{3}\right)-5xy^{4}-\left(-5x^{4}y\right)-5x^{3}y^{2}-4y^{3}x^{2}-\left(-2y^{5}x\right)-y^{6}-\left(-y^{3}x^{3}\right)-y^{4}x^{2}+12x^{3}y^{2}-6x^{2}y^{4}+3xy^{5}-3x^{4}y^{2}+3x^{3}y^{3}$$
- step5: Remove the parentheses:
  $$12x^{5}-6x^{4}y^{2}+3x^{3}y^{3}-3x^{6}+3x^{5}y+8x^{4}y-4x^{3}y^{3}+2x^{2}y^{4}-2x^{5}y+2x^{4}y^{2}-20x^{3}y+10x^{2}y^{3}-5xy^{4}+5x^{4}y-5x^{3}y^{2}-4y^{3}x^{2}+2y^{5}x-y^{6}+y^{3}x^{3}-y^{4}x^{2}+12x^{3}y^{2}-6x^{2}y^{4}+3xy^{5}-3x^{4}y^{2}+3x^{3}y^{3}$$
- step6: Calculate:
  $$12x^{5}-7x^{4}y^{2}+3x^{3}y^{3}-3x^{6}+x^{5}y+13x^{4}y-5x^{2}y^{4}-20x^{3}y+6x^{2}y^{3}-5xy^{4}+7x^{3}y^{2}+5y^{5}x-y^{6}$$
Aquí están los resultados de las operaciones de polinomios solicitadas:

### a) $$(A + B) \times (C - D)$$

Primero, sumamos $$A$$ y $$B$$:
$$
A + B = (x^{3}+2 x^{2} y-5 x y+y^{3}) + (2 x^{3}+3 x y^{2}-2 y^{3}) = 3x^{3}+2 x^{2} y-5 x y+y^{3}+3 x y^{2}-2 y^{3}
$$
Esto se simplifica a:
$$
3x^{3}+2 x^{2} y-5 x y+3 x y^{2}+y^{3}-2 y^{3} = 3x^{3}+2 x^{2} y-5 x y+3 x y^{2}-y^{3}
$$

Luego, restamos $$D$$ de $$C$$:
$$
C - D = (4 x^{2}-x y^{2}+y^{3}) - (x^{3}-x^{2} y+x y^{2}) = 4 x^{2}-x y^{2}+y^{3}-x^{3}+x^{2} y-x y^{2}
$$
Esto se simplifica a:
$$
-x^{3}+5x^{2}-2xy^{2}+y^{3}
$$

Finalmente, multiplicamos los resultados:
$$
(A + B) \times (C - D) = 12x^{5}-7x^{4}y^{2}+3x^{3}y^{3}-3x^{6}+x^{5}y+13x^{4}y-5x^{2}y^{4}-20x^{3}y+6x^{2}y^{3}-5xy^{4}+7x^{3}y^{2}+5y^{5}x-y^{6}
$$

### b) $$(D \times A) - (C + B)$$

Primero, multiplicamos $$D$$ y $$A$$:
$$
D \times A = (x^{3}-x^{2} y+x y^{2}) \times (x^{3}+2 x^{2} y-5 x y+y^{3}) = x^{6}+x^{5}y-5x^{4}y+3x^{3}y^{3}-x^{4}y^{2}+5x^{3}y^{2}-x^{2}y^{4}-5x^{2}y^{3}+xy^{5}-4x^{2}-2xy^{2}+y^{3}-2x^{3}
$$

Luego, sumamos $$C$$ y $$B$$:
$$
C + B = (4 x^{2}-x y^{2}+y^{3}) + (2 x^{3}+3 x y^{2}-2 y^{3}) = 2x^{3}+4x^{2}+2xy^{2}+y^{3}
$$

Finalmente, restamos los resultados:
$$
(D \times A) - (C + B) = x^{6}+x^{5}y-5x^{4}y+3x^{3}y^{3}-x^{4}y^{2}+5x^{3}y^{2}-x^{2}y^{4}-5x^{2}y^{3}+xy^{5}-4x^{2}-2xy^{2}+y^{3}-2x^{3}
$$

### c) $$\frac{(A - C) \times B}{D}$$

Primero, restamos $$C$$ de $$A$$:
$$
A - C = (x^{3}+2 x^{2} y-5 x y+y^{3}) - (4 x^{2}-x y^{2}+y^{3}) = x^{3}-2x^{2}+xy^{2}-5xy
$$

Luego, multiplicamos el resultado por $$B$$:
$$
(A - C) \times B = (x^{3}-2x^{2}+xy^{2}-5xy) \times (2 x^{3}+3 x y^{2}-2 y^{3}) = 2x^{5}+5x^{3}y^{2}+4x^{2}y^{3}+4x^{4}y-xy^{4}-10yx^{3}-7y^{3}x+10y^{4}-8x^{4}-12x^{2}y^{2}-2y^{5}
$$

Finalmente, dividimos por $$D$$:
$$
\frac{(A - C) \times B}{D} = \frac{2x^{5}+5x^{3}y^{2}+4x^{2}y^{3}+4x^{4}y-xy^{4}-10yx^{3}-7y^{3}x+10y^{4}-8x^{4}-12x^{2}y^{2}-2y^{5}}{x^{2}-xy+y^{2}}
$$

### d) $$(C - A) + (B \times D)$$

Primero, restamos $$A$$ de $$C$$:
$$
C - A = (4 x^{2}-x y^{2}+y^{3}) - (x^{3}+2 x^{2} y-5 x y+y^{3}) = 4x^{2}-x^{3}-2x^{2}y+5xy
$$

Luego, multiplicamos $$B$$ y $$D$$:
$$
B \times D = (2 x^{3}+3 x y^{2}-2 y^{3}) \times (x^{3}-x^{2} y+x y^{2}) = 2x^{6}-2x^{5}y+5x^{4}y^{2}-5x^{3}y^{3}+5x^{2}y^{4}-2y^{5}x
$$

Finalmente, sumamos los resultados:
$$
(C - A) + (B \times D) = 4x^{2}-xy^{2}-x^{3}-2x^{2}y+5xy+2x^{6}-2x^{5}y+5x^{4}y^{2}-5x^{3}y^{3}+5x^{2}y^{4}-2y^{5}x
$$

Estos son los resultados finales de las operaciones de polinomios.
Here are the results of the polynomial operations: a) $$(A + B) \times (C - D) = 12x^{5} - 7x^{4}y^{2} + 3x^{3}y^{3} - 3x^{6} + x^{5}y + 13x^{4}y - 5x^{2}y^{4} - 20x^{3}y + 6x^{2}y^{3} - 5xy^{4} + 7x^{3}y^{2} + 5y^{5}x - y^{6}$$ b) $$(D \times A) - (C + B) = x^{6} + x^{5}y - 5x^{4}y + 3x^{3}y^{3} - x^{4}y^{2} + 5x^{3}y^{2} - x^{2}y^{4} - 5x^{2}y^{3} + xy^{5} - 4x^{2} - 2xy^{2} + y^{3} - 2x^{3}$$ c) $$\frac{(A - C) \times B}{D} = \frac{2x^{5} + 5x^{3}y^{2} + 4x^{2}y^{3} + 4x^{4}y - xy^{4} - 10yx^{3} - 7y^{3}x + 10y^{4} - 8x^{4} - 12x^{2}y^{2} - 2y^{5}}{x^{2} - xy + y^{2}}$$ d) $$(C - A) + (B \times D) = 4x^{2} - xy^{2} - x^{3} - 2x^{2}y + 5xy + 2x^{6} - 2x^{5}y + 5x^{4}y^{2} - 5x^{3}y^{3} + 5x^{2}y^{4} - 2y^{5}x$$

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