In this chapter we will learn to factorize polynomial by regrouping the individual entities.

In the end solutions are also provided for further clarity.

## Regroup and factorize

Sometimes the entity given in the polynomial are arranged in such a way that it gets difficult to factorize them.

In this method we will first** rearrange different entities so that they have common elements between them** and then do the **factorization using group method**.

To factorize by regrouping, follow the below steps;

(a) Rearrange the entities such that after forming group we have common factor.

(b) Now arrange the entities in different group.

(c) Factorize each group individually

(d) Take out the common element and solve the remaining expressions.

I hope you understood the above process. Let us see some solved examples for further clarity.**Example 01**

Factorize the below polynomial;

\mathtt{\Longrightarrow \ ax^{2} +by^{2} +bx^{2} +ay^{2}} **Solution**

Note that the first and last two elements do not have any common element between them. So factorization is not possible.

Here we have to **rearrange the elements to form group with common elements**.

\mathtt{ax^{2} \ \&\ \ bx^{2}} both have \mathtt{x^{2}}

\mathtt{ay^{2} \ \&\ \ by^{2}} both have \mathtt{y^{2}}

So we **arrange the above elements adjacent to each other.**

Regrouping the polynomial, we get;

\mathtt{\Longrightarrow \ ax^{2} +by^{2} +bx^{2} +ay^{2}}\\\ \\ \mathtt{\Longrightarrow \ ax^{2} +bx^{2} +ay^{2} +by^{2}}

Now **form group of first and last two elements and then factorize.**

\mathtt{\Longrightarrow \ \left( ax^{2} +bx^{2}\right) +\left( ay^{2} +by^{2}\right)}\\\ \\ \mathtt{\Longrightarrow \ x^{2}( a+b) \ +\ y^{2}( a+b)}\\\ \\ \mathtt{\Longrightarrow \ ( a+b) \ \left( x^{2} +y^{2}\right)}

Hence, we get two factors \mathtt{( a+b) \ \left( x^{2} +y^{2}\right)} after the factorization.

**Example 02**

Rearrange and factorize the polynomial

\mathtt{\Longrightarrow \ xy+mn+xm+yn} **Solution**

Rearrange the entities so that we have common element between the adjacent entities.

\mathtt{\Longrightarrow \ xy+mn+xm+yn}\\\ \\ \mathtt{\Longrightarrow \ xy\ +xm\ +yn+mn}

Now form the group of first two and last two entities. Form the group and factorize.

\mathtt{\Longrightarrow \ ( xy\ +xm) \ +( yn+mn)}\\\ \\ \mathtt{\Longrightarrow \ x( y+m) +n( y+m)}\\\ \\ \mathtt{\Longrightarrow \ ( y+m)( x+n)}

Hence, we got the factors (y+m) and (x+n) as final solution.

**Example 03**

Factorize the polynomial

\mathtt{\Longrightarrow \ 5xy\ +\ 3x\ +\ 5y\ +3} **Solution****5xy & 5y** both have common element **5y**.**3x & 3** have common element **3**.

Arranging the polynomial to have entities with common elements adjacent to each other.

\mathtt{\Longrightarrow \ 5xy\ +\ 5y\ +\ 3x\ +\ 3} **Form group of first and last two elements.**

\mathtt{\Longrightarrow \ ( 5xy\ +\ 5y) \ +\ ( 3x\ +\ 3)}\\\ \\ \mathtt{\Longrightarrow \ 5y( x+1) \ +3\ ( x+1)}\\\ \\ \mathtt{\Longrightarrow \ ( x+1)( 5y+3)}

Hence, **we simplified the polynomial to (x+1)(5y+3)**.

**Example 04**

Rearrange and factorize the polynomial.

\mathtt{\Longrightarrow \ 7xy-50-35y+10x} **Solution****7xy & 35y** have common factor **7y.****10x & 50** have common factor **10**.

Arranging these entity adjacent to each other and then factorizing using group method.

\mathtt{\Longrightarrow \ 7xy-50-35y+10x}\\\ \\ \mathtt{\Longrightarrow \ ( 7xy\ -\ 35y) +\ ( 10x\ -50)}\\\ \\ \mathtt{\Longrightarrow \ 7y( x-5) \ +\ 10\ ( x\ -\ 5)}\\\ \\ \mathtt{\Longrightarrow \ ( x-5) \ ( 7y\ +\ 10)}

Hence, we simplified the polynomial to **(x – 5).(7y+10)**

**Example 05**

\mathtt{\Longrightarrow z\ –\ 21\ +\ 21xy\ –\ xyz}

**Solution****21 & 21xy** have common factor **21**.**z & xyz** have common factor **z**.

Arranging these entities adjacent to each other then factorizing using group method.

\mathtt{\Longrightarrow \ ( -21\ +\ 21xy) \ +( z-xyz)}\\\ \\ \mathtt{\Longrightarrow \ -21\ ( 1-xy) \ +z( 1-xy)}\\\ \\ \mathtt{\Longrightarrow \ ( 1-xy) \ ( -21\ +\ z)}\\\ \\ \mathtt{\Longrightarrow \ ( 1-xy) \ ( z-21)}

Hence, the polynomial is reduced to** (1 – xy) (z – 21)**