Algebra 2- Find the least common multiple of the following expression

Question

10a^2b, 25ab^3, 5b^4

Answer 1

10*(a^2)*b factored is 2*5*a*a*b

25*a*(b^3) factored is 5*5*a*b*b*b

5*(b^4) factored is 5*b*b*b*b

 

The least common multiple must include all factors. However, we only need each factor to the highest degree they show up in any of the polynomials. This means, for example, since we have a term with a^2, our least common multiple must have a^2, but it doesn't need to be higher than that. Also, we have one polynomial with a factor of 5, and another with a factor of 5*5=25. Therefore, our least common multiple must have the higher degree factor, 5^2 (or 25). Lastly, all three terms have at least one b, and the highest degree of b is 4. Therefore, our LCM must have b^4.

 

So,

LCM = 2*5*5*a*a*b*b*b*b = 50*(a^2)*(b^4).

Answer 2

Hi Billy, work this type of problem in parts. One part are the coefficients, one part is the "a" variable, and the last part is the "b" variable.

 

The coefficients:

 

The three coefficients are 10, 25, and 5. They are all multiples of 5. What is the largest multiple of 5 that can be found in the three coefficients? Answer: 5.

 

5 will divide into 10 and 25. 10 will not divide into 5 and 25. 25 will not divide into 10 and 5.

 

"a" variable:

 

The "a" variable does not occur in all three terms. Therefore in cannot be used when finding the common multiple.

 

"b" variable:

 

The "b" variable can be found in all three terms. The exponents for the "b" variable is 1 in the first term (it is implied), 3 in the second term, and 4 in the last term.

 

What is the largest exponent that can be taken from all three terms? Answer: 1.

 

You cannot take more than one "b" from the first term.

 

Answer:

 

Putting the three parts back together again gives a solution of: 5b out of each term.

 

What is left after the 5b is removed:

 

10a²b becomes 2a² when the 5b is removed.

 

15ab³ becomes 3ab² when the 5b is removed.

 

5b⁴ becomes b³ when the 5b is removed.

 

Questions?

Answer 3

Hello,

Let's first look constant (or number in front of the variables).

the three constants are: 10, 25, 5

 

What number is common among all three of the constants?

 

That number will be least common multiple for the constants.

 

Next let's look at the variable component.

The variable components for each section are: a^2*b, ab^3, b^4

 

What letter is common among all variables?

 

You can remove that letter from the all three components and have your finalized least common multiple.

 

example: 10a^4*b, 5a^3*b^2, 5b^4

the letter b is in each expression, so I can pull it out so that it looks like this:

10A^4*b, 5b*a^3*b, 5b*b^3

 

lastly the constant can be factored by 5 leading to the last result:

 

(5b)(2a^4), (5b)(a^3*b), (5b)(b^3)

 

Apply this to your question and you can factor to the least common multiple, which from the example was:

(5b).

 

Thank you,

Hunter Haddad