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Factoring Polynomials example ( Enter your problem )
  1. Formula
  2. Examples

2. Examples


Type-1 (Taking common) Examples...
(1) `ax + a + 2x + 2`
`=(ax + a) + (2x + 2)`
`=a(x + 1) + 2(x + 1)`
`=(x + 1)(a + 2)`

Type-2 (Difference of squares) Examples...
(1) `25x^2 - 36`
`=(5x)^2 - (6)^2`
`=(5x - 6)(5x + 6)`

Type-3 (Sum and Difference of cubes) Examples...
(1) `x^3 + 27`
`=(x)^3 + (3)^3`
`=(x + 3)(x^2 - (x)(3) + (3)^2)`
`=(x + 3)(x^2 - 3x + 9)`

Type-4 (Whole square of a bionomial) Examples...
(1) `4x^2 + 12xy + 9y^2`
`=(2x)^2 + 2(2x)(3y) + (3y)^2`
`=(2x + 3y)^2`

Type-5 (Splitting the middle term of a Quadratic Equation) Examples...
(1) `x^2 + 10x + 24`
`=x^2 + 4x + 6x + 24`
`=x(x + 4) + 6(x + 4)`
`=(x + 4)(x + 6)`

Type-6 (Whole square of a trinomial) Examples...
(1) `4x^2 + y^2 + 1 + 4xy + 4x + 2y`
`=(2x)^2 + (y)^2 + (1)^2 + 2(2x)(y) + 2(2x)(1) + 2(y)(1)`
`=(2x + y + 1)^2`

Type-7 (Factorization with the help of factor theorem) Examples...
(1) `x^3 - 3x^2 - 6x + 8`
Here `p(x)=x^3 - 3x^2 - 6x + 8`
sum of coefficient of all the terms of `p(x) = 1 - 3 - 6 + 8 = 0`
`:.` `(x-1)` is a factor of `p(x)`.
Now, `p(x) = x^3 - 3x^2 - 6x + 8`
`=x^3 - x^2 - 2x^2 + 2x - 8x + 8`
`=x^2(x - 1) - 2x(x - 1) - 8(x - 1)`
`=(x - 1)(x^2 - 2x - 8)`
`=(x - 1)(x - 4)(x + 2)`

Type-8 Cyclic Expressions Examples...
(1) `a^2(b - c) + b^2(c - a) + c^2(a - b)`
`= a^2b - a^2c + b^2c - b^2a + c^2a - c^2b`
`= a^2b - a^2c + cb^2 - ab^2 + ac^2 - bc^2`
`= a^2b - a^2c - ab^2 + ac^2 + b^2c - bc^2`
`= a^2(b - c) - a(b^2 + c^2) + bc(b - c)`
`= a^2(b - c) - a(b - c)(b + c) + bc(b - c)`
`= (b - c)(a^2 - a(b + c) + bc)`
`= (b - c)(a^2 - ab - ac + bc)`
`= (b - c)(a(a - b) - c(a - b))`
`= (b - c)(a - b)(a - c)`
`= -(a - b)(b - c)(c - a)`



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