Find the H.C.F of:
a) [tex]$2 x^2(x+2)(x-2)$[/tex] and [tex]$4 x(x+2)(x+3)$[/tex]

The highest common factor of the expressions 2 x 2 ( x + 2 ) ( x − 2 ) and 4 x ( x + 2 ) ( x + 3 ) is 2 x ( x + 2 ) . This is found by factorizing both expressions and identifying the common factors. By multiplying the common factors raised to their lowest powers, we arrive at the H.C.F.
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Answered by Anonymous | 2025-07-04

Factorize both expressions: 2 x 2 ( x + 2 ) ( x − 2 ) = 2 ⋅ x ⋅ x ⋅ ( x + 2 ) ⋅ ( x − 2 ) and 4 x ( x + 2 ) ( x + 3 ) = 2 ⋅ 2 ⋅ x ⋅ ( x + 2 ) ⋅ ( x + 3 ) .
Identify the common factors: 2, x, and (x+2).
Multiply the common factors with the lowest power: 2 ⋅ x ⋅ ( x + 2 ) .
Simplify the expression: 2 x ( x + 2 ) ​ .

Explanation

Understanding the Problem We are asked to find the highest common factor (H.C.F) of two algebraic expressions. The expressions are 2 x 2 ( x + 2 ) ( x − 2 ) and 4 x ( x + 2 ) ( x + 3 ) . The H.C.F is the product of the common factors of the expressions, each raised to the lowest power it appears in either expression.

Factorizing the Expressions First, let's factorize both expressions completely:


Expression 1: 2 x 2 ( x + 2 ) ( x − 2 ) = 2 ⋅ x ⋅ x ⋅ ( x + 2 ) ⋅ ( x − 2 )
Expression 2: 4 x ( x + 2 ) ( x + 3 ) = 2 ⋅ 2 ⋅ x ⋅ ( x + 2 ) ⋅ ( x + 3 )

Identifying Common Factors Now, let's identify the common factors in both expressions. Both expressions have the factors 2, x, and (x+2) in common.

The lowest power of 2 that appears in both expressions is 2 1 = 2 .
The lowest power of x that appears in both expressions is x 1 = x .
The lowest power of (x+2) that appears in both expressions is ( x + 2 ) 1 = ( x + 2 ) .

Calculating the H.C.F The H.C.F is the product of these common factors:

H.C.F = 2 ⋅ x ⋅ ( x + 2 )

Simplifying the H.C.F Finally, let's simplify the H.C.F:

H.C.F = 2 x ( x + 2 )
Examples
Understanding H.C.F is crucial in various real-life scenarios, such as simplifying fractions, determining the largest size of tiles to cover a floor without cutting, or optimizing resource allocation. For instance, if you have two pieces of land with areas represented by the given expressions, the H.C.F would represent the largest common area that can be equally divided from both pieces. This concept helps in efficient planning and distribution of resources.

Answered by GinnyAnswer | 2025-07-04