How do you find the LCD for #2 /(2x+6)# , #15 /(2x^2 + 12x +18)#?

Answer 1

#=>2(x+3)^2#

You find the LCD for polynomial expressions the same way you would with just numbers.

For example, if we want to find the LCD for #1/3# and #1/6#, we look for the smallest multiple of #3# and #6# that is shared. In this case, it happens to be #6#. We can always find a common denominator by taking the product: #3xx6 = 18#, but this is not always the LCD. In this case, because #6# is a multiple of #3#, we can do better!

For polynomials, we apply the same logic. We try to find the smallest multiple of both polynomial denominators.

We are given two fractions which can be simplified first by factoring:

#2/(2x+6) = 2/(2(x+3)) = color(blue)(1/(x+3))#
#15/(2x^2+12x+18)=15/(2(x^2+6x+9))=color(blue)(15/(2(x+3)^2))#
We notice that the second fraction has an extra power of #(x+3)# in its denominator and an extra multiple of #2#. This second denominator is our least common denominator, because we already have #x+3# in the first fraction. There is nothing in the first fraction that isn't in the second.
To see that the second fraction denominator is the LCD, let's substitute #a equiv (x+3)#.

So we have these two fractions:

#1/a, 15/(2a^2)#
You can always find a common denominator by multiplying the two denominators. In this case we would get a common denominator of #2a^3#.
But like the example I gave first, this isn't necessarily the LCD. In this case (for the same reasons I showed with numbers), #a^2# is a multiple of #a#, so #a^2# is sufficient. Now, the only thing that differs between the fractions is a factor of #2# in the denominator, so we need to keep that as well. So we've determined that the LCD for these fractions is #2a^2#, which is precisely #color(green)(2(x+3)^2)#.
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Answer 2

To find the least common denominator (LCD) for the fractions 2/(2x+6) and 15/(2x^2 + 12x + 18), we need to factor the denominators and identify the common factors.

The denominator of the first fraction, 2x+6, can be factored as 2(x+3).

The denominator of the second fraction, 2x^2 + 12x + 18, can be factored as 2(x^2 + 6x + 9), which further simplifies to 2(x+3)(x+3).

The common factors in the denominators are 2 and (x+3).

Therefore, the LCD for the fractions is 2(x+3)(x+3).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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