Which of the following represents the factorization of the trinomial below?
[tex]x^2+24 x+144[/tex]
A. [tex](x-12)(x-2)[/tex]
B. [tex](x-12)^2[/tex]
C. [tex](x+12)(x-12)[/tex]
D. [tex](x+12)^2[/tex]
Answer (1)
Recognize the trinomial as a perfect square: x 2 + 24 x + 144 .
Identify the square root of the constant term: 144 = 12 .
Express the trinomial as a squared binomial: ( x + 12 ) 2 .
The correct factorization is ( x + 12 ) 2 .
Explanation
Understanding the Problem We are given the trinomial x 2 + 24 x + 144 and asked to find its factorization from the given options.
Recognizing Perfect Square Trinomial We can recognize that the given trinomial is in the form of a perfect square trinomial, which can be factored as ( x + a ) 2 = x 2 + 2 a x + a 2 or ( x − a ) 2 = x 2 − 2 a x + a 2 .
Identifying the Value of 'a' In our case, we have x 2 + 24 x + 144 . We notice that 144 = 1 2 2 . So, we can try to express the trinomial as ( x + 12 ) 2 .
Expanding the Potential Factorization Expanding ( x + 12 ) 2 , we get x 2 + 2 ( 12 ) x + 1 2 2 = x 2 + 24 x + 144 , which matches the given trinomial.
Concluding the Factorization Therefore, the factorization of the trinomial x 2 + 24 x + 144 is ( x + 12 ) 2 .
Selecting the Correct Option Comparing our result with the given options, we find that option D, ( x + 12 ) 2 , is the correct factorization.
Examples
Factoring trinomials is a fundamental skill in algebra and is used in many real-world applications. For example, engineers use factoring to simplify complex equations when designing structures, ensuring stability and efficiency. Similarly, economists use factoring to analyze market trends and predict future economic behavior. Understanding factoring helps in simplifying and solving problems across various fields.
