Productos Notables

• pre-calculus
• Tertiary Education

Productos Notables

Los productos notables son patrones comunes en las multiplicaciones algebráicas que permiten resolverlos fácilmente.

Producto algebráico

Para resolver una multiplicación algebráica se multiplica cada término

$$(x+3)(x^2+x-2)$$

$$(x\cdot x^2)+(x\cdot x)+(x\cdot (-2))+(3\cdot x^2)+(3\cdot x)+(3\cdot (-2))$$

$$x^3+x^2-2x+3x^2+3x-6$$

$$x^3+4x^2+x-6$$

Binomio al cuadrado

$$(a\pm b{)}^2$$

$$(a\pm b{)}^2=(a\pm b)(a\pm b)$$

$$(a\pm b{)}^2=(a\cdot a)+(a\cdot (\pm b))+((\pm b)\cdot a)+((\pm b)\cdot (\pm b))$$

$$(a\pm b{)}^2=a^2\pm ab\pm ab+b^2$$

$$(a\pm b{)}^2=a^2\pm 2ab+b^2$$

Binomio al cuadrado

$$(a\pm b{)}^2=a^2\pm 2ab+b^2$$

$$(x+3{)}^2=x^2+2\cdot x\cdot 3+(3{)}^2=x^2+6x+9$$

$$(2x-5{)}^2=(2x{)}^2-2\cdot (2x)\cdot 5+(3{)}^2=4x^2-20x+25$$

$$(x^2-y{)}^2=(x^2{)}^2-2\cdot x^2\cdot y+y^2=x^4-2x^{2}y+y^2$$

Resuelve:

$$(3x-4{)}^2=$$

1. 3x​2​​−24x+16
2. 9x​2​​−24x+16
3. 9x​2​​+24x+16
4. 3x​2​​+24x+16

Binomio al cubo

$$(a\pm b{)}^3$$

$$(a\pm b{)}^3=(a\pm b)(a^2\pm 2ab+b^2)$$

$$(a\pm b{)}^3=a^3\pm 2a^{2}b+a{b}^2\pm a^{2}b+2a{b}^2\pm b^3$$

$$(a\pm b{)}^3=a^3\pm 3a^{2}b+3a{b}^2\pm b^3$$

Binomio al Cubo

$$(a\pm b{)}^3=a^3\pm 3a^{2}b+3a{b}^2\pm b^3$$

$$(x+3{)}^3=x^3+3\cdot x^2\cdot 3+3\cdot x\cdot (3{)}^2+3^3=x^3+9x^2+27x+27$$

$$(2x-5{)}^3=(2x{)}^3-3(2x{)}^2(5)+3(2x)(5{)}^2-(5{)}^3=8x^3-60x^2+150x-125$$

$$(x^2-y{)}^3=(x^2{)}^3-3(x^2{)}^2(y)+3(x^2)(y{)}^2-y^3=x^6-3x^{4}y+3x^2{y}^2-y^3$$

Resuelve:

$$(3x-4{)}^3=$$

1. 27x​3​​−108x​2​​+144x−64
2. 27x​3​​+108x​2​​+144x+64
3. 3x​3​​−36x​2​​+144x−64
4. 3x​3​​+36x​2​​+144x+64

Binomios conjugados

$$(a+b)(a-b)$$

$$(a+b)(a-b)=(a\cdot a)+(a\cdot b)+(a\cdot (-b))+(b\cdot (-b))$$

$$(a+b)(a-b)=a^2+ab-ab-b^2$$

$$(a+b)(a-b)=a^2-b^2$$

Binomios conjugados

$$(a+b)(a-b)=a^2-b^2$$

$$(x+3)(x-3)=x^2-(3^2)=x^2-9$$

$$(2x-5)(2x+5)=4x^2-25$$

$$(x^2+y)(x^2-y)=x^4-y^2$$

Resuelve:

$$(3x-4)(3x+4)=$$

1. 9x​2​​−24x+16
2. 3x​2​​−24x+16
3. 9x​2​​−16
4. 3x​2​​−16

Resuelve:

$$(5x+5{)}^3=$$

1. 125x​3​​+375x​2​​+375x+125
2. 5x3+75x2+375x+125
3. 5x​3​​+50x​2​​+50x+25
4. 25x​2​​+50x+25

Resuelve:

$$(3x^4-1{)}^2=$$

1. 3x​8​​+6x​4​​+1
2. 3x8−6x4+1
3. 9x​8​​−6x​4​​+1
4. 9x​8​​+6x​4​​+1

Resuelve:

$$(3x^2-y)(3x^2+y)=$$

1. 3x​2​​+y​2​​
2. 9x4+y2
3. 3x​4​​−y​2​​
4. 9x​4​​−y​2​​