In Algebra, Polynomial Factoring is the process of expressing a polynomial equation as a product of two or more similar polynomials. Each polynomial as a product of the given polynomial equation. For example (x - 2) and (x + 2) are the factors of x^{2} - 4

x^{2} - 4 = (x - 2) (x + 2)

Thus factoring is mainly used for solving or simplifying polynomial expressions. The process of polynomial factoring is also called as the resolution into factors.

**How to Find Factoring of Polynomial:**

There are three methods (common factors, grouping terms and using formulas) are used to find polynomial factors in common

**Factoring Polynomials by Common Factor:**

First we have to find the common factor for the given polynomial equation. If a given algebraic polynomial equation **A** contains common factor **B**, then divide each term of equation A by common factor B and you will get an expression C. Now B and C are the factors of A and it can be expressed as A = B x C

**Example**: Find the factors for the polynomial equation 7x^{5}y^{3} + 4 x^{2}y^{3} + 10 xy^{3}?
__Solution:__

In the above equation xy^{3} is common factor.

7x^{5}y^{3} + 4 x^{2}y^{3} + 10 xy^{3} = xy^{3} (7x^{5}y^{3}/xy^{3} + 4 x^{2}y^{3}/xy^{3} + 10 xy^{3}/xy^{3})

7x^{5}y^{3} + 4 x^{2}y^{3} + 10 xy^{3} = xy^{3} (7x^{4}+4x+10)

**Try Yourself**

Find the polynomial Factors for the below equations

1. 9x - 3y

2. 4x^{3} - 8x^{2} + 16x

3. p^{5} + 4p

4. 6a^{5}b^{5} + 3a^{2}b^{3} + 14ab^{3}

5. 7ab - 21a^{2}b^{22}

**Factoring Polynomials by Grouping terms**

If the Polynomial Equation don't have common factors then group the terms. By grouping terms in a appropriate manner you can get a common factor.
**Example**: Find polynomial factors for the below equation

x^{2} +x - 2xy - 2y
__Solution:__

x^{2} +x - 2xy - 2y = x^{2} - 2xy +x - 2y

Take X as common for first two terms

= x(x - 2y) + (x - 2y)

= (x - 2y) (x + 1)

So the factors for x^{2} +x - 2xy - 2y are (x - 2y) and (x + 1)

**Try Yourself**

Find the polynomial Factors for the below equations

1. ab - 2c - cb + 2a

2. a^{3} - 2a^{2} - 2a + 4

3. a^{3} - a^{2} - ba + b

4. 2x^{3} - x^{2} + 2x - 1

5. 8a^{3} + 4a^{2} + 4a + 2

**Polynomial Factors by Formulas**

In some cases, the below factorization formulas can be used to resolving a polynomial into factors.

1. (x + y)^{2} = x^{2} + y^{2} + 2xy

2. (x - y)^{2} = x^{2} + y^{2} - 2xy

3. (x + y) (x - y) = x^{2} - y^{2}

4. (x + y) (x^{2} - xy + y^{2}) = x^{3} - y^{3}

5. (x - y) ( x^{2} + xy + y^{2}) = x^{3} - y^{3}

6. (x + y)^{2} = x^{3} + y^{3} + 3x^{2}y + 3y^{2}x = x^{3} + y^{3} + 3xy (x + y)

7. (x - y)^{3} = x^{3} - y^{3} - 3x^{2}y + 3y^{2}x = x^{3} - y^{3} - 3xy (x - y)

8. (x + y + z)^{2} = x^{2} + y^{2} + z^{2} + 2xy + 2yz + 2zx

9. (x + y + z)(x^{2} + y^{2} + z^{2} - xy - yz - zx) = x^{3} + y^{3} + z^{3} - 3xyz

**Examples Problems**

1. Find factors of the equation 4a^{2} + 12ab + 9b^{2}
__Solution:__

The given polynomial equation can be written as follows

4a^{2} + 12ab + 9b^{2} = (2a)^{2} + 2 (2a)(3b) + (3b)^{2}

The above equation can be simplified by using (x + y)^{2}

= (2a + 3b)^{2}

2. Find factors of the equation 125a^{3} + 64b^{3}
__Solution:__

125a^{3} = (5a)^{3}

64b^{3} = (4b)^{3}

lets consider x = 5a and y = 4b

125a^{3} + 64b^{3} = (5a)^{3} + (4b)^{3}

x^{3} + y^{3} = (x + y)(x^{2} +y^{2} - xy)

= (5a + 4b)[(5a)^{2} + (4b)^{2} - (5a)(4b)]

= (5a + 4b)[25a^{2} + 16b^{2} - 20ab]

3. Factorize 6(a-1)^{2}b - 5(a - 1)b^{2} - 6b^{3}
__Solution:__

put x= a-1 in the above equation,

6(a-1)^{2}b - 5(a - 1)b^{2} - 6b^{3} = 6(x)^{2}b - 5(x)b^{2} - 6b^{3}

= b(6x^{2} - 5bx - 6b^{2})

Here we find 6 X -6 = -36 = -9 X 4 and -9 + 4 = -5

= b(6x^{2} - 9bx + 4bx - 6b^{2})

= b[3x(2x - 3b) + 2b(2x - 3b)]

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

= b[2(a-1) - 3b] [3(a - 1) + 2b]

= b[(2a - 3b -2)(3a + 2b - 3)]

There are quadratic with integer coefficients which can not be factored with integer coefficients

**Try Yourself:**

Find the polynomial factors for the below equations

1. x^{4} + x^{2}y^{2} + y^{4}

2. 4a^{2} + 20ab + 25b^{2} - 10a - 25b

3. 27a^{3} + b^{3} + 27a^{2}b + 9ab^{2}

4. a^{3} - b^{3} + 1 + 3ab

5. 3√3a^{3}b^{3} + 27c^{3}

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