# FACTOR POLYNOMIALS BY GROUPING

To factor polynomial using grouping method,

i) Divide the polynomial into groups.

ii) Find the greatest common factor is every group.

iii) That is the value we have to take out from all the terms in the particular group.

Factor the following polynomial by grouping.

Problem 1 :

x4 - 2x³ + 3x² - 6x

Solution :

= x4 - 2x³ + 3x² - 6x

Dividing the given polynomial into two groups,

Group 1 :

= x4 - 2x³

GCF of x4 and 2x³ is x3Factoring x3, we get

= x3(x - 2)

Group 2 :

= 3x² - 6x

GCF of 3x² - 6x is 3x. Factoring 3x, we get

= 3x(x - 2)

= x3(x - 2) + 3x(x - 2)

= (x - 2) (x3 + 3x)

= (x - 2) ⋅ x (x2 + 3)

= x (x - 2)(x2 + 3)

So, factors are x (x - 2)(x2 + 3).

Problem 2 :

x5 - x4 - 2x3 + 2x²

Solution :

= x5 - x4 - 2x3 + 2x²

Factoring

= x²(x³ - x² - 2x + 2)

= x²[x²(x - 1) - 2(x - 1)]

= x²(x - 1) (x² - 2)

So, factors are x²(x - 1) (x² - 2).

Problem 3 :

x4 - 3x³ - 5x² + 15x

Solution :

= x4 - 3x³ - 5x² + 15x

= x(x³ - 3x² - 5x + 15)

= x[x²(x - 3) - 5(x - 3)]

= x(x² - 5) (x - 3)

So, factors are x(x² - 5) (x - 3).

Problem 4 :

x² (x - 1) - 9(x - 1)

Solution :

x² (x - 1) - 9(x - 1)

(x - 1) (x² - 9)

= (x - 1) (x² - 3²)

Using algebraic identity,

a² - b² = (a + b) (a - b)

(x - 1) (x + 3) (x - 3)

So, factors are (x - 1) (x + 3) (x - 3).

Problem 5 :

x³ - 2x² - 16x + 32

Solution :

= x³ - 2x² - 16x + 32

= (x³ - 2x²) + (-16x + 32)

= x²(x - 2) - 16 (x - 2)

= (x - 2) (x² - 16)

= (x - 2) (x² - 4²)

Using algebraic identity,

a² - b² = (a + b) (a - b)

= (x - 2) (x + 4) (x - 4)

So, factors are (x - 2) (x + 4) (x - 4).

Problem 6 :

4x²(x - 3) - 25(x - 3)

Solution :

= 4x²(x - 3) - 25(x - 3)

= (x - 3) (4x² - 25)

= (x - 3) [(2x)² - 5²]

Using algebraic identity,

a² - b² = (a + b) (a - b)

= (x - 3) (2x + 5) (2x - 5)

So, factors are (x - 3) (2x + 5) (2x - 5).

Problem 7 :

x³ + 4x² - 36x - 144

Solution :

= x³ + 4x² - 36x - 144

= (x³ + 4x²) + (- 36x - 144)

= x²(x + 4) - 36(x + 4)

= (x + 4) (x² - 36)

= (x + 4) (x² - 6²)

Using algebraic identity,

a² - b² = (a + b) (a - b)

= (x + 4) (x + 6) (x - 6)

So, factors are (x + 4) (x + 6) (x - 6).

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