$27 a^3+108 a^2 b+144 a b^2+64 b^3$

The expression 27 a 3 + 108 a 2 b + 144 a b 2 + 64 b 3 can be simplified to ( 3 a + 4 b ) 3 by recognizing it as a binomial expansion. This is achieved by identifying appropriate terms for x and y . After verifying the terms, the final expression is ( 3 a + 4 b ) 3 .
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Answered by Anonymous | 2025-07-05

The problem involves simplifying the expression 27 a 3 + 108 a 2 b + 144 a b 2 + 64 b 3 .

Recognize the expression as a binomial expansion.
Identify x = 3 a and y = 4 b .
Rewrite the expression as ( 3 a + 4 b ) 3 .
The simplified expression is ( 3 a + 4 b ) 3 ​ .

Explanation

Analyzing the Problem We are given the expression 27 a 3 + 108 a 2 b + 144 a b 2 + 64 b 3 . Our goal is to simplify this expression. It appears to be a cubic polynomial in terms of a and b .

Recognizing the Pattern We recognize that the given expression might be a binomial expansion of the form ( x + y ) 3 = x 3 + 3 x 2 y + 3 x y 2 + y 3 . We need to identify x and y such that x 3 = 27 a 3 and y 3 = 64 b 3 .

Finding x and y Taking the cube root of 27 a 3 and 64 b 3 , we get x = 3 27 a 3 ​ = 3 a and y = 3 64 b 3 ​ = 4 b . So, we have x = 3 a and y = 4 b .

Verifying the Middle Terms Now, let's check if the middle terms match the binomial expansion with x = 3 a and y = 4 b . We have 3 x 2 y = 3 ( 3 a ) 2 ( 4 b ) = 3 ( 9 a 2 ) ( 4 b ) = 108 a 2 b and 3 x y 2 = 3 ( 3 a ) ( 4 b ) 2 = 3 ( 3 a ) ( 16 b 2 ) = 144 a b 2 . These match the given expression.

Rewriting the Expression Since all terms match the binomial expansion, we can rewrite the expression as ( 3 a + 4 b ) 3 . Therefore, 27 a 3 + 108 a 2 b + 144 a b 2 + 64 b 3 = ( 3 a + 4 b ) 3 .


Examples
Polynomial factorization is a fundamental concept in algebra and has practical applications in various fields. For instance, in engineering, simplifying complex polynomial expressions can help in designing efficient systems. Consider a scenario where the volume of a container is expressed as a polynomial. By factoring this polynomial, engineers can determine the dimensions of the container that satisfy specific volume requirements. This ensures optimal use of materials and space, leading to cost-effective and sustainable designs. Factoring polynomials also plays a crucial role in cryptography, where it is used to create secure encryption algorithms.

Answered by GinnyAnswer | 2025-07-05