Question

Fatore as expressões :

$a) 5a^{4} -3a^{3}b-45a^{2}b^{2}+27ab^{3}\\\\b)a^{4}-a^{2}+16\\c)x^{5}+y^{5}-xy^{4}-x^{4}y\\d)(x^{2}+y^{2}-5)^{2} - 4(xy-2)^{2}\\e)[(1+x^{2})(1+x^{2})-4xy]^{2}-4[x(1+y^{2})-y(1+x^{2})]^{2}$

a)a(a+3b)(a+3b)(5a-3b)

b)(a^{2}+4)^{2})-9a^{2}

c)(x^{2}+y^{2})(x+y)(x-y)^{2}

d)(x-y-3)(x-y+3)(x+y-1)(x+y+1)

e)(1-x)^{2}(1+x)^{2}(1-y)^{2}(1+y)^{2}



não estou conseguindo e nem sabendo chegar ao gabarito

Answer 1

Explicação passo-a-passo:

Letra a)

5(a²)² - 3a³b - 45a²b² + 27ab³ =

5(a²)² - 45a²b² - 3a³b + 27ab³ =

5a²a² - 45a²b² - 3a³b + 27ab³ =

5a²(a² - 9b²) - 3ab(a² - 9b²) =

(a² - 9b²)[(5a²) - 3ab] =

(a² - 9b²)(5a² - 3ab) =

[a² - (3b)²][a(5a - 3b)] =

[(a + 3b)(a - 3b)][a(5a - 3b)] =

a(a + 3b)(a - 3b)(5a - 3b).

Letra b)

a²a² - a² + 16 =

a²a² + (- a²) + 16 =

a²a² + (8a² - 9a²) + 16 =

(a²)² + 8a² + 16 - 9a² =

[(a²)² + 8a² + 16] - 9a² =

(a² + 4)² - (3a)² =

[(a² + 4) + (3a)][(a² + 4) - (3a)] =

(a² + 4 + 3a)(a² + 4 - 3a).

Letra c)

x(x²)² + y(y²)² - x(y²)² - y(x²)² =

x(x²)² - x(y²)² + y(y²)² - y(x²)² =

x[(x²)² - (y²)²] + y[(y²)² - (x²)²] =

x[(x²)² - (y²)²] + y(- 1)[(x²)² - (y²)²] =

x[(x²)² - (y²)²] - y[(x²)² - (y²)²] =

(x - y)[(x²)² - (y²)²] =

(x - y)[(x² + y²)(x² - y²)] =

(x - y)(x² + y²)(x + y)(x - y) =

(x² + y²)(x + y)(x - y)².

Letra d)

(x² + y² - 5)² - 4(xy - 2)² =

(x² + y² - 5)² - 2²(xy - 2)² =

(x² + y² - 5)² - (2xy - 4)² =

[(x² + y² - 5) + (2xy - 4)][(x² + y² - 5) - (2xy - 4)] =

[(x² + 2xy + y²) - 5 - 4][(x² - 2xy + y²) - 5 + 4] =

[(x + y)² - 9][(x - y)² - 1] =

[(x + y)² - 3²][(x - y)² - 1²] =

[(x + y) + 3][(x + y) - 3][(x - y) + 1][(x - y) - 1] =

(x + y + 3)(x + y - 3)(x - y + 1)(x - y - 1).

Letra e)

— Primeira Resolução

[(1 + x²)(1 + y²) - 4xy]² - 2²[x(1 + y²) - y(1 + x²)]² =

[(1 + x²)(1 + y²) - 4xy]² - [2x(1 + y²) - 2y(1 + x²)]² =

{[(1 + x²)(1 + y²) - 4xy] + [2x(1 + y²) - 2y(1 + x²)]}{[(1 + x²)(1 + y²) - 4xy] - [2x(1 + y²) - 2y(1 + x²)]} =

[(1 + x²)(1 + y²) - 4xy + 2x(1 + y²) - 2y(1 + x²)][(1 + x²)(1 + y²) - 4xy + 2y(1 + x²) - 2x(1 + y²)] =

[(1 + x²)(1 + y² - 2y) + 2x(1 + y² - 2y)][(1 + x²)(1 + y² + 2y) - 2x(1 + y² + 2y)] =

[(x² + 2x + 1)(y² - 2y + 1)][(x² - 2x + 1)(y² + 2y + 1)] =

(x² + x + x + 1)(y² - y - y + 1)(x² - x - x + 1)(y² + y + y + 1) =

[x(x + 1) + (x + 1)][y(y - 1) - (y - 1)][x(x - 1) - (x - 1)][y(y + 1) + (y + 1)] =

[(x + 1)(x + 1)][(y - 1)(y - 1)][(x - 1)(x - 1)][(y + 1)(y + 1)] =

(x + 1)²(y - 1)²(x - 1)²(y + 1)² =

(1 - x)²(1 + x)²(1 - y)²(1 + y)².

— Segunda Resolução

[(1 + x²)(1 + y²) - 4xy]² - 2²[x(1 + y²) - y(1 + x²)]² =

(1 + x²)²(1 + y²)² - 2(1 + x²)(1 + y² m)4xy + 4²x²y² - 2²[x²(1 + y²)² - 2x(1 + y²)y(1 + x²) + y²(1 + x²)²] =

(1 + x²)²(1 + y²)² - 8xy(1 + x²)(1 + y²) + 16x²y² - 4x²(1 + y²)² + 8xy(1 + x²)(1 + y²) - 4y²(1 + x²)² =

(1 + x²)²(1 + y²)² + [8xy(1 + x²)(1 + y²) - 8xy(1 + x²)(1 + y²)] + 16x²y² - 4x²(1 + y²)² - 4y²(1 + x²)² =

(1 + x²)²(1 + y²)² + 0 + 16x²y² - 4x²(1 + y²)² - 4y²(1 + x²)² =

(1 + x²)²(1 + y²)² - 4x²(1 + y²)² - 4y²(1 + x²)² + 16x²y² =

(1 + y²)²[(1 + x²)² - 4x²] - 4y²[(1 + x²) - 4x²] =

[(1 + y²)² - 4y²][(1 + x²)² - 4x²] =

[(x² + 1)² - (2x)²][(y² + 1)² - (2y)²] =

[(x² + 1) + (2x)][(x² + 1) - 2x][(y² + 1) + (2y)][(y² + 1) - (2y)] =

(x² + 2x + 1)(x² - 2x + 1)(y² + 2y + 1)(y² - 2y + 1) =

(x² + x + x + 1)(x² - x - x + 1)(y² + y + y + 1)(y² - y - y + 1) =

[x(x + 1) + (x + 1)][x(x - 1) - (x - 1)][y(y + 1) + (y + 1)][y(y - 1) - (y - 1)] =

[(x + 1)(x + 1)][(x - 1)(x - 1)][(y + 1)(y + 1)][(y - 1)(y - 1)] =

(x + 1)²(x - 1)²(y + 1)²(y - 1)².

Abraços!