Find the lowest common multiple of [tex]$x^2+2 x+1$[/tex] and [tex]$x^2-6 x+9$[/tex].
A. [tex]$(x+1)(x-3)$[/tex]
B. [tex]$(x+1)^2(x-3)^2$[/tex]
C. [tex]$(x-1)^2(x+3)^2$[/tex]
D. [tex]$(x+1)(x-1)(x+3)(x-3)$[/tex]

Factor the first polynomial: x 2 + 2 x + 1 = ( x + 1 ) 2 .
Factor the second polynomial: x 2 − 6 x + 9 = ( x − 3 ) 2 .
Identify the distinct factors and their highest powers: ( x + 1 ) 2 and ( x − 3 ) 2 .
The LCM is the product of these highest powers: ( x + 1 ) 2 ( x − 3 ) 2 ​ .

Explanation

Understanding the Problem We are given two polynomials, x 2 + 2 x + 1 and x 2 − 6 x + 9 , and we need to find their lowest common multiple (LCM). The LCM is the smallest polynomial that is divisible by both given polynomials.

Factoring the Polynomials First, we factorize the given polynomials.


The first polynomial is x 2 + 2 x + 1 . This is a perfect square trinomial, which can be factored as follows: x 2 + 2 x + 1 = ( x + 1 ) ( x + 1 ) = ( x + 1 ) 2
The second polynomial is x 2 − 6 x + 9 . This is also a perfect square trinomial, which can be factored as follows: x 2 − 6 x + 9 = ( x − 3 ) ( x − 3 ) = ( x − 3 ) 2

Finding the LCM Now that we have the factored forms of the polynomials, we can find the LCM. The LCM is the product of the highest powers of all distinct factors present in either polynomial.

The distinct factors are ( x + 1 ) and ( x − 3 ) . The highest power of ( x + 1 ) is 2, and the highest power of ( x − 3 ) is 2. Therefore, the LCM is: ( x + 1 ) 2 ( x − 3 ) 2

Final Answer Comparing our result with the given options, we see that option B matches our result.

Therefore, the lowest common multiple of x 2 + 2 x + 1 and x 2 − 6 x + 9 is ( x + 1 ) 2 ( x − 3 ) 2 .
Examples
Understanding LCM is crucial in many real-world scenarios. For example, when scheduling events that occur at different intervals, like planning when two buses on different routes will meet again at a station. If one bus comes every 12 minutes and another every 18 minutes, finding the LCM of 12 and 18 (which is 36) tells you they'll meet every 36 minutes. This concept extends to more complex scheduling, resource allocation, and even music theory, where understanding harmonic intervals involves finding common multiples of frequencies.

Answered by GinnyAnswer | 2025-07-08

The lowest common multiple of the polynomials x 2 + 2 x + 1 and x 2 − 6 x + 9 is ( x + 1 ) 2 ( x − 3 ) 2 , which corresponds to option B.
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Answered by Anonymous | 2025-07-23