Matemáticas
Descomposición factorial
Casos de Factores
Ejercicio 106
Miscelanea sobre los 10 casos de descomposición en factores
5a² + a R/ a(5a + 1)
m² + 2mx + x²
m x R/ (m + x)²
a² + a - ab - b
a(a + 1) -b(a + 1) R/ (a - b) (a + 1)
x² + 36 R/ (x + 6) (x - 6)
9x² - 6xy + y²
3x y
(3x - y)(3x - y) R/ (3x - y)²
x² - 3x - 4
(x + 4)(x + 1) R/ (x + 4) (x + 1)
6x² - x - 2
(2x + 1)(3x - 2) R/ (2x + 1) (3x - 2)
1 + x³ R/ (1 + x) (1 - x + x²)
27a³ - 1 R/ (3a - 1) (9a² + 3a + 1)
x5 + m5
(x5 + m5) / (x + m)
x4 - mx3 + m²x² - m3x + m4 R/ (x + m) (x4 - mx3 + m²x² - m3x + m4)
a³ - 3a²b + 5ab² R/ a (a² - 3ab + 5b²)
2xy - 6y + x2 - 32 R/ (x - 3) ( 3ab + 5b²)
1 - 4b + 4b² R/ (1 - 2b)²
4x4 + 3x²y² + y4 R/ (2x² + xy + y²)(2x² - xy + y²)
x8 - 6x4y4 + y8 R/ (x4 + 2x²y² - y4) (x4 - 2x²y² - y4)
a² - a - 30 R/ (a - 6)(a + 5)
15 m² + 11m - 14 R/ (3m - 2) (5m + 7)
a6 + 1 R/ (a² + 1) (a4 - a² + 1)
8m3 - 27y6 R/ (2m - 3y²)(4m² + 6my² + 9y4)
16a² - 24 ab + 9b² R/ (4a - 3b)²
1 + a7 R/ (1 + a)(1 - a + a² - a³ + a4 - a5 + a6)
8a³ - 12a² + 6ª - 1 R/ (2a - a)³
1 - m² R/ (1 + m) (1 - m)
x4 + x2 - 21 R/ (x² + 7)(x² - 3)
125 a6 + 1 R/ (5a² + 1)(25a4 - 5a2 + 1)
a² + 2ab + b² - m² R/ (a + b + m)(a + b - m)
8a² + 16 a³b - 24 a²b² R/ 8a²b(1 + 2a - 3b)
x5 - x4 + x - 1 R/ (x4 + 1)( x - 1)
6x² + 19x - 20 R/ (6x - 5)(x + 4)
25x4 - 81y2 R/ (5x² + 9y)(5x² - 9y)
1 - m³ R/ (1 - m)(1 + m + m²)
x² - a² + 2xy + y² + 2ab -b² R/ (x + y + a -b)( x + y - a + b)
21m5n - 7m4n² + 7m³n³ - 7m²n R/ 7m²n(3m³ - m²n + mn² - 1)
a(x + 1) - b(x + 1) + c(x + 1)
(x + 1)(a - b + c) R/ (x + 1)(a - b + c)
4 + 4(x - y) + (x - y)²
4 + 4x - 4y + (x - y)²
1 - a²b4
(1 - ab²)(1 + ab²) R/ (1 - ab²)(1 + ab²)
b² + 12ab + 36a²
b 6ª
(6a + b)(6a + b)
(6ª + b)² R/ (6ª + b)²
x6 + 4x³ - 77
(x³ - 7)(x³ + 11) R/ (x³ - 7)(x³ + 11)
15x4 - 17x² - 4
(15x²) - 17(15x²) - 60
(15x² - 20) (15x² + 3)/ 3 x 5
(3x² - 4)(5x² + 1) R/ (3x² - 4)(5x² + 1)
1 + (a - 3b)³
1 + a³ - (3b)³
(1 + a - 3b)(1 - a + 3b + a² - 2a(3b) - 9b²) R/ (1 + a - 3b)(1 - a + 3b + a² - 6ab - 9b²)
x4 + x² + 25
(x² + x + 5)(x² - x + 5) R/ (x² + x + 5)(x² - x + 5)
a8 - 28a4 + 36
(a4 - 4a² + 6)( a4 - 4a² - 6) R/ (a4 - 4a² + 6)( a4 - 4a² - 6)
343 + 8a³
8a³ + 343
(2a + 7)[(2a)² - 2(7a) + (7)²]
(2a + 7)(4a² - 14a + 49) R/ (2a + 7) (4a² - 14a + 49)
12a²bx - 15a²by
(3a²b)(4x - 5y) R/ (3a²b)(4x - 5y)
x² + 2xy - 15y²
(x - 3y)(x + 5y) R/ (x - 3y)(x + 5y)
6am - 4an - 2 n + 3m
81a6 - 4b2c8
16 -(2a + b)²
16 - 4a² + b²
(4 + 2a + b) (4 - 2a - b) R/ (4 + 2a + b)( 4 - 2a - b)
20 - x -x²
-x² - x + 20 (-1)
x² + x - 20
(x + 5)(x - 4) R/ (x + 5)(x - 4)
n² + n - 42
(n + 7)(n - 6) R/ (n + 7)(n - 6)
a² - d² + n² - c² - 2an - 2cd R/ (a - n + c + d)(a - n - c - d)
1 + 216x9
(1 + 6x³)(1 - 2(1)(6x³) + (6x³)²)
(1 + 6x³)(1 - 12x³ + 36x6) R/ (1 + 6x³)(1 - 12x³ + 36x6)
x³ - 64
(x - 4)(x² - 2(x)(4) + (4)²)
(x - 4)(x² - 8x + 16) R/ (x - 4)(x² - 8x + 16)
x³ - 64x4
x³(1 - 64x) R/ x³(1 - 64x)
18ax5y3 - 36x4y3 - 54x2y8
18x²y³(ax³ - 2x² - 3y5) R/ 18x²y³(ax³ - 2x² - 3y5)
49a²b² - 14ab + 1
7ab 1
(7ab - 1)² R/ (7ab - 1)²
(x + 1)² - 81
x² + 2x + 1 - 81
(x - 8)(x + 10) R/ (x - 8)(x + 10)
a² - (b + c)²
a² - b² + c²
(a - b - c)(a + b + c) R/ (a + b + c)(a - b - c)
(m + n)² - 6 (m + n) + 9
m+n 3
(m + n - 3) (m + n - 3) R/(m + n - 3)²
7x² + 31x - 20
7(7x²) - 7(31x) - 20 (7)
7x² - 31(7x) - 140
(7x + 35)(7x - 4)/1 x 7
(x + 5)(7x - 4) R/ (x + 5)(7x - 4)
9a³ + 63a - 45a²
9a³ - 45a² + 63ª
9a (a² - 5ª + 7) R/ 9a (a² - 5a + 7)
ax + a - x -1
(a - 1)(x + 1) R/ (a - 1)(x + 1)
81x4 + 25y2 - 90x²y
9x² 5y
(9x² - 5y)(9x² - 5y)
(9x² - 5y)² R/ (9x² - 5y)²
1 - 25b² + b4
1 -5b+b²
(1 - 5b - b²)(1 + 5b - b²) R/ - 5b - b²)(1 + 5b - b²)
m4 + m2n2 + n4
m² mn n²
(m² + mn + n²)(m² - mn - n²) R/ (m² + mn + n²)(m² - mn - n²)
c4 - 4d4
c² 2d²
(c² + 2d²)( c² - 2d²) R/ (c² + 2d²)( c² - 2d²)
15x4 -15x³ + 20x²
5x²(3x² - 3x + 4) R/ 5x²(3x² - 3x + 4)
a² - x² - a - x
(a + x)(a - x - 1) R/ (a + x)(a - x - 1)
x4 - 8x² - 240
(x² - 20 )(x² + 12) 8 240 2
4 120 2
2 60 2
1 30 2
15 3
5 5
1 R/ (x² + 12)(x² - 20)
6m4 + 7m² - 20
(2m² + 12 )(3m² - 4) R/ (2m² + 12)(3m² - 4)
9n² + 4a² - 12an
9n² - 12 an + 4a²
3n 2a
(3n - 2a)² R/ (3n - 2a)²
2x² + 2
2(x² + 1) R/ 2(x² + 1)
7a (x + y -1) - 3b(x + y - 1)
(7a - 3b)(x + y - 1) R/ (x + y - 1)(7a - 3b)
x² + 3x - 18
(x + 6)(x - 3) R/ (x + 6)(x - 3)
(a + m)² - (b + n)²
(a + m)(a + m) + (-b - n)(b + n) R/ (a + m + b + n)(a + m - b - n)
x³ + 6x²y + 12xy² + 8y³
(x + 2y)(x + 2y)² R/ (x + 2y)³
8a² - 22a - 21
(4a + 3)(2ª - 7) R/ (4a + 3)(2a -7)
1 + 18ab + 81a²b²
(9ab)² + 2(9ab) + 1
(9ab + 1)(9ab + 1) R/ (1 + 9ab)²
4a6 - 1
2a³ 1
(2a³ + 1)(2a³ - 1) R/ (2a³ + 1)( 2a³ - 1)
x6 - 4x³ - 480
(x³ - 24)(x³ + 20) R/ (x³ - 24)(x³ + 20)
ax - bx + b - a - by + ay
x(a - b)+y(a - b)-1(a - b) R/ (a - b)(x + y - 1)
6am - 3m -2a +1
3m(2a - 1) - 1(2a - 1) R/ (3m - 1)(2a - 1)
15 + 14x - 8x²
-8x² + 14x + 15
(4x + 3)(-2x + 5) R/ (3 + 4x)(5 - 2x)
a10 - a8 + a6 + a4
a4(a6 - a4 + a² + 1) R/ a4(a6 - a 4 + a² +1)
2x(a - 1)- a + 1
2ax - 2x - a + 1
a(2x - 1)-1(2x - 1) R/ (2x - 1)(a - 1)
(m + n)(m - n) + 3n(m - n)
m - n[m + n + 3n]
(m - n)(m + 4n) R/ (m + 4n)(m - n)
a² - b³ + 2b³x² - 2a²x²
a² - 2a²x² - b³ + 2b³x²
a²(1 - 2x²) - b³(1 - 2x²)
(a² - b³)(1 - 2x²) R/ (a² - b³)(1- 2x²)
2am - 3b - c - cm - 3bm + 2a
m(2a - c - 3b)+ 1(2a - 3b - c) R/ (m + 1)(2a - 3b - c)
x² - 2/3x + 1/9
(x - 1/3)(x + 1/3) R/ (x - 1/3)²
4a2n - b4n
(2ªn + b2n) (2ªn - b2n) R/ (2ªn + b2n) (2ªn - b2n)
81x² - (a + x)²
()() R/ (10x + a)(8x - a)
a² + 9 - 6a - 16x² R/ (a - 3 + 4x)( a - 3 - 4x)
9a² - x² - 4 + 4x R/ (3a + x - 2) (3a - x + 2)
9x² - y² + 3x - y R/ (3x - y)(3x + y + 1)
x² - x - 72 R/ (x - 9)(x + 8)
36a4 - 120a²b² + 49b4 R/ (6a² + 6ab - 7b²)(6a² - 6ab - 7b²)
a² - m² - 9n² - 6mn + 4ab + 4b² R/ (a + 2b + m + 3n) (a + 2b - m - 3n)
1 - 4/9a8 R/ (1 + 2/3a4)(1 - 2/3a4)
81a8 + 64b12 R/ (9a4 + 12a²b³ + 8b6) (9a4 - 12a²b³ + 8b6)
49x² - 77x + 30
(7x)² - 11(7x) + 30 R/ (7x - 5)(7x - 6)
x² - 2abx - 35a²b² R/ (x - 7ab)(x + 5ab)
125x³ - 225x² + 135x - 27
(125x³ - 27)-(225x² - 135x)
(5x - 3)(25x² + 15x + 9) - 45x(5x - 3)
(5x - 3)(25x² + 15x + 9 - 45x) factor comun (5x - 3)
(5x - 3)(25x² - 30x - 9)
(5x - 3)(5x - 3)² R/ (5X - 3)³
(a - 2)² - (a + 3)²
[(a - 2) - (a - 3)][(a - 2) - (a + 3)]
(a - 2 - a - 3)(a - 2 + (a + 3))
(-5)(a - 2 + a + 3) R/ -5 (2a + 1)
4a²m + 12 a²n - 5bm - 15bn
4a²(m + 3n) - 5b(m + 3n)
(m + 3n)²(4a² - 5b) R/ (4a² - 5b)(m + 3n)
1 + 6x³ + 9x6
9x6 + 6x³ + 1
(3x3)² + 2(3x³) + 1
(3x³ + 1)(3x³ + 1) R/ (1 + 3x³)²
a4 + 3a²b - 40b²
(a²)² + 3b(a²) - 40b² R/ (a² - 5b)(a² + 8b)
m³ + 8a³x³
(m + 2ax)(m - 2ax)² R/ (m+2ax)(m² - 2amx + 4a²x²)
1 - 9x² + 24xy - 16y²
(-1) 9x² - 24 xy + 16y² - 1
(3x²) - 8y(3x) + 16y²) - 1
(3x - 4y)(3x - 4y) - 1 R/ (1 + 3x - 4y)(1 - 3x + 4y)
1 + 11x + 24x²
(24x)² + 11(24x) + 24
(24x + 8)(24x + 3)/8*3 R/ (3x + 1)(8x + 1)
9x²y³ - 27x³y³ - 9x5y3
9x²y³(1 - 3x - x³) R/ 9x²y³(1 - 3x - x³)
(a² + b² - c²)² - 9x²y²
[(a² + b² + c²)- 3xy][(a² + b² - c²+ 3xy] R/ (a² + b² - c² + 3xy) (a² + b² - c² - 3xy)
8(a + 1)³ - 1
(2a + 2)³ - 1
(2a + 1)(2ª + 2)² + (2ª + 2)(1) + (1)²
(2a + 1)(4a² + 4 + 2ª + 2 +1)
(2a + 1)4a² + 2(2a)(2) + 2²
(2a + 1)4a² + 8ª + 4 + 2a + 2 + 1 R/ (2a + 1)(4a² + 10a +7)
100x4y6 - 121m4
(10x²y³)² - 11m² R/ (10x²y³ + 11m²)(10x²y³ - 11m²)
(a² + 1)² + 5(a² + 1) - 24
(a² + 1 + 8)( a² + 1 - 3) R/ (a² + 9)(a² - 2)
1 + 1000x6
(1 + 10x²)(1 - 10x² + 100x4) R/ (1 + 10x²)(1 - 10x² + 100x4)
49a² - x² -9y² + 6xy
49a² (-x² - 9y² + 6xy)
49a² - (x² - 6xy + 9y²)
49a² - [(x - 3y)(x - 3y)]
(49a²)-(x - 3y)²
(7a)² - (x - 3y)² R/ (7a + x - 3y)(7a -x + 3y)
x4 - y² + 4x² + 4 - 4yz - 4z²
x4 + 4x² + 4 - y² - 4yz - 4z²
(x4 + 4x² + 4) - (y² + 4yz + 4z²)
[(x² + 2)(x² + 2)][(y + 2z)(y + 2z)]
(x² + 2)² - (y + 2z)² R/ (x² + 2 + y + 2z)( x² + 2 - y - 2z)
a³ - 64 R/ (a - 4)(a² + 4ª + 16)
a5 + x5 R/ (a + x)(a4 - a3x + a²x² - ax³ + x4)
a6 - 3a3b - 54b² R/ (a³ + 6b)(a³ - 9b)
165 + 4x - x²
-(x² - 4x - 165)
-[(x - 15)(x + 11)] R/ (11 + x)(15 - x)
a4 + a2 +1
(a² + a)² + 1
(a² + a + 1)(a² - a + 1) R/ (a² + a + 1)(a² - a + 1)
x²/4 - y6/81
(x/2 - y³/9)( x/2 + y³/9) R/ (x/2 + y³/9)(x/2 - y³/9)
16x² + 8xy/5 + y²/25
4x² + (4x)(2y/5) + y²/25
(4x + y/5)( 4x + y/5) R/ (4x + y/5)²
a4b4 + 4a2b2 - 96
(a²b² + 12)( a²b² - 8) R/ (a²b² + 12)(a²b² - 8)
8a²x + 7y +21by - 7ay -8a³x + 24a²bx
8a²x - 24a² bx - 7ay
8a²x (1 + 3b - a) + 7y (3b - a + 1) R/ (8a²x + 7y)(1 - a + 3b)
x4 + 11x² - 390
(x² + 26)(x² - 15) R/ (x² + 26)( x² - 15)
7 + 33m - 10m²
-10m² + 33m + 7 R/ (1 + 5m)(7 - 2m)
4(a + b)² - 9(c + d)²
(2(a + b))² - (3(c + d))²
(2a + 2b)² - (3c + 3d)²
[(2a + 2b) + (3c + 3d)][(2a + 2b) - (3c + 3d)] R/ (2a + 2b + 3c + 3d)(2ª + 2b - 3c - 3d)
729 - 125x³y12
(9 - 5xy4)(81 + 45xy4 + 25 x²y8) R/ (9 - 5xy4)(81 + 45xy4 + 25x²y8)
(x + y)² + x + y
x² + 2xy + y² + x + y
x² + x + 2xy + y + y²
x y
(x + y)(x + y + 1) R/ (x + y)(x + y + 1)
4 - (a² + b²) + 2ab
a² - b² + 2ab + 4
(a² + 2ab - b²) + 4
(a + 1b)(a + 1b) + 4
(2 + a - b)(2 - a + b) R/ (2 + a - b)(2 - a + b)
x³ - y³ + x - y
x³ + x - y + y³
x(x² + 1) - y(y² + 1)
(x - y)(x² + xy + y² + 1) R/ (x - y)(x² + xy + y² + 1)
a² - b² + a³ - b³
a³ + a² - b² - b³
a(a² + a)-b(b - b²)
(a - b)(a² + ab + b² + a + b) R/ (a - b)(a² + ab + b² + a + b)
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Enviado por: | Tomcat |
Idioma: | castellano |
País: | El Salvador |