Inicio › Matemáticas 3 › Productos notables

Matemáticas 3 · Tema 11

Productos notables

Ejemplos

(A+B)(C+D)=AC+AD+BC+BD(A + B)(C + D) = AC + AD + BC + BD
(5x−2y)(6x−3y)=30x2−15xy−12yx+6y2\begin{aligned} (5x - 2y)(6x - 3y) &= 30x^2 - 15xy - 12yx + 6y^2 \end{aligned}
30x2−27xy+6y230x^2 - 27xy + 6y^2
(a+b)2=a2+2ab+b2(a + b)^2 = a^2 + 2ab + b^2
(3x2−yx4)2=(3x2)2+2(3x2)(−yx4)+(−yx4)2\begin{aligned} (3x^2 - yx^4)^2 &= (3x^2)^2 + 2(3x^2)(-yx^4) + (-yx^4)^2 \end{aligned}
9x4−6x6y+y2x89x^4 - 6x^6y + y^2x^8
(a+b)3=a3+3a2b+3ab2+b3(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3
(5x−2)3=(5x)3+3(5x)2(−2)+3(5x)(−2)2+(−2)3=125x3+3(25x2)(−2)+3(5x)(4)−8\begin{aligned} (5x - 2)^3 &= (5x)^3 + 3(5x)^2(-2) + 3(5x)(-2)^2 + (-2)^3 \\ &= 125x^3 + 3(25x^2)(-2) + 3(5x)(4) - 8 \end{aligned}
125x3−150x2+60x−8125x^3 - 150x^2 + 60x - 8
(a+b)(a−b)=a2−b2(a + b)(a - b) = a^2 - b^2
(4x3−5y)(4x3+5y)=(4x3)2−(5y)2\begin{aligned} (4x^3 - 5y)(4x^3 + 5y) &= (4x^3)^2 - (5y)^2 \end{aligned}
16x6−25y216x^6 - 25y^2

Ejercicios

Bloque 1
(3x−2y)(5xy−4xy)(3x - 2y)(5xy - 4xy)
(4x2−3yx)2(4x^2 - 3yx)^2
(4x−1)3(4x - 1)^3
(6x2−2y)(6x2+2y)(6x^2 - 2y)(6x^2 + 2y)
Bloque 2
(2x+3y)(4xy−6xy)(2x + 3y)(4xy - 6xy)
(2x2+5yx)2(2x^2 + 5yx)^2
(6x2−3)3(6x^2 - 3)^3
(3x2+5y)(3x2−5y)(3x^2 + 5y)(3x^2 - 5y)

Respuestas

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Bloque 1

(3x−2y)(5xy−4xy)(3x - 2y)(5xy - 4xy)
=15x2y−12x2y−10xy2+8xy2\begin{aligned} &= 15x^2y - 12x^2y - 10xy^2 + 8xy^2 \end{aligned}
3x2y−2xy23x^2y - 2xy^2
(4x2−3yx)2(4x^2 - 3yx)^2
=(4x2)2+2(4x2)(−3yx)+(−3yx)2\begin{aligned} &= (4x^2)^2 + 2(4x^2)(-3yx) + (-3yx)^2 \end{aligned}
16x4−24x3y+9y2x216x^4 - 24x^3y + 9y^2x^2
(4x−1)3(4x - 1)^3
=(4x)3+3(4x)2(−1)+3(4x)(−1)2+(−1)3=64x3+3(16x2)(−1)+3(4x)(1)−1\begin{aligned} &= (4x)^3 + 3(4x)^2(-1) + 3(4x)(-1)^2 + (-1)^3 \\ &= 64x^3 + 3(16x^2)(-1) + 3(4x)(1) - 1 \end{aligned}
64x3−48x2+12x−164x^3 - 48x^2 + 12x - 1
(6x2−2y)(6x2+2y)(6x^2 - 2y)(6x^2 + 2y)
=(6x2)2−(2y)2\begin{aligned} &= (6x^2)^2 - (2y)^2 \end{aligned}
36x4−4y236x^4 - 4y^2

Bloque 2

(2x+3y)(4xy−6xy)(2x + 3y)(4xy - 6xy)
=8x2y−12x2y+12xy2−18xy2\begin{aligned} &= 8x^2y - 12x^2y + 12xy^2 - 18xy^2 \end{aligned}
−4x2y−6xy2-4x^2y - 6xy^2
(2x2+5yx)2(2x^2 + 5yx)^2
=(2x2)2+2(2x2)(5yx)+(5yx)2\begin{aligned} &= (2x^2)^2 + 2(2x^2)(5yx) + (5yx)^2 \end{aligned}
4x4+20x3y+25y2x24x^4 + 20x^3y + 25y^2x^2
(6x2−3)3(6x^2 - 3)^3
=(6x2)3+3(6x2)2(−3)+3(6x2)(−3)2+(−3)3=216x6+3(36x4)(−3)+3(6x2)(9)−27\begin{aligned} &= (6x^2)^3 + 3(6x^2)^2(-3) + 3(6x^2)(-3)^2 + (-3)^3 \\ &= 216x^6 + 3(36x^4)(-3) + 3(6x^2)(9) - 27 \end{aligned}
216x6−324x4+162x2−27216x^6 - 324x^4 + 162x^2 - 27
(3x2+5y)(3x2−5y)(3x^2 + 5y)(3x^2 - 5y)
=(3x2)2−(5y)2\begin{aligned} &= (3x^2)^2 - (5y)^2 \end{aligned}
9x4−25y29x^4 - 25y^2

Ejercicios extra

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Bloque 1
(2x+5y)(3x−4y)(2x + 5y)(3x - 4y)
(3x2−2y)2(3x^2 - 2y)^2
(2x+3)3(2x + 3)^3
(5x2−4y)(5x2+4y)(5x^2 - 4y)(5x^2 + 4y)
(4x−7y)(2x+3y)(4x - 7y)(2x + 3y)
Bloque 2
(6x3+y2)2(6x^3 + y^2)^2
(3x−2)3(3x - 2)^3
(7x2+2y3)(7x2−2y3)(7x^2 + 2y^3)(7x^2 - 2y^3)
(x2−5xy)2(x^2 - 5xy)^2
(2x2+y)(5x−3y)(2x^2 + y)(5x - 3y)
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