Factorization of Algebraic Expressions

Factorization is breaking down a mathematical expression into smaller components or factors, which when multiplied, gives the original expression.

Factorization helps us understand how the expression works and makes solving equations easier.

An algebraic expression is a math phrase made up of numbers, letters, variables, and any basic operations (+ − × ÷). It represents a value, but since it has no equal sign, it doesn't define the exact value.

Variables like x or y stand for unknown values, constants are fixed numbers, and operators suggest what to do, like add, subtract, multiply, or divide. 

Factoring is rewriting an algebraic expression as a product of its simple terms. We break them into factors that multiply to give the original expression.

For example, 2y + 6, these both share a common factor of 2. 2y = 2y, 6 = 23. If we factor out 2, we get 2y+6 = 2(y+3). 2 is multiplied by (y+3), which gives back the original expression when expanded.

Taking out a common factor

Factoring out the greatest common factor means taking out the biggest number and variable combination that’s common to every term.

First, we will find the HCF of all part numbers and variables.

Now divide every term by it.

Write this outside the parentheses.

For example, 6x2 + 9x  HCF is 3x , we get 3x(x+3)

Factor by grouping

Factor by grouping means grouping the terms so that every group shares common factors.

Separate into 2 groups.

Now factor in every group.

Take out what is common to both groups.

For example, 3m2+3mn+2m+2n

Group into two parts and factor each.

(3m2+3mn) + (2m+2n) = 3m(m+n) +2 (m+n)

As (m+n) is common, factor this out, we get (m+n)(3m+2)

Difference of squares

a²- b² = (a+b) (a-b)

For example, 25a² - 64b²  that's (5a)² - (8b)²

We get (5a + 8b) (5a - 8b)

To factorize algebraic expressions, we identify common patterns like squares or cubes.

• 4x2+12x+9
   
• 4x2 is (2x)2, and 9 is 32
   
• Middle term 12x equals 2(2x)3  . It matches the pattern a²+b², where a = 2x, b = 3.
   
• So it factors to (2x+3)²
   

Factorization of algebraic expressions is useful in many fields, like architecture, finance, and economics. Some real-life applications of the factorization of algebraic expressions are mentioned below.

• Optimizing Area in Architecture: Architects apply this to optimize the complete area in measurements for layout planning. For example, an architect needs to design a garden with area A = x2 + 5x + 6 square unit. To find out the length and width   A = (x+2)(x+3). 
   
• Cost Analysis in Business: Factorization helps in identifying how costs change with production and identifying cost drivers. For example, a business has a cost function. Suppose the cost function be C(x) = x²+ 5x +6. Factorize to get C(x) = (x+2)(x+3), which reveals components of fixed cost and variable cost. 
   
• Engineering Force Distribution: This helps in revealing symmetric force points as well as failure thresholds in a structured way. For example, a mechanical engineer needs to know the stress at the breaking point, and the force function is F(x) = x² - 8. Factorized to know the stress at the breaking point, which is F(x)=(x-8)(x+8). Which is disclosing the critical points where stress may be on top.
   
• Profit Boost in Economics: This is useful in production where profit becomes positive or can be zero.  A profit function is p(x) = x²-6x +4. Factors as p(x)=(x-3+5)(x-3-5).
   
• Solving Quadratic Paths in Sports:  This is useful to find when the ball hits the ground and when it reaches its peak height. For example, football follows the trajectory h(t)=5t2+10t. Factorize gives h(t)=5t(t+2) ball hits the ground when h(t)=0 time t=0 and t=-2.