Assuming that 3u + v neq 0, simplify (54u^2 v + 18uv^2)(9u + 3v).

 May 9, 2023


To simplify the expression (54u^2v + 18uv^2)(9u + 3v), we can use the distributive property of multiplication.

First, let's distribute 9u to each term inside the first parentheses:  TellPopeyes

9u * 54u^2v = 486u^3v
9u * 18uv^2 = 162u^2v^2

Next, let's distribute 3v to each term inside the first parentheses:

3v * 54u^2v = 162uv^3
3v * 18uv^2 = 54uv^3

Now, we can combine like terms:

486u^3v + 162u^2v^2 + 162uv^3 + 54uv^3

To simplify this further, we can group the terms with the same variables:

(486u^3v + 162u^2v^2) + (162uv^3 + 54uv^3)

Inside each group, we can factor out common terms:

486u^3v + 162u^2v^2 = 162u^2v(3u + v)
162uv^3 + 54uv^3 = 216uv^3(3u + v)

Now, we have:

162u^2v(3u + v) + 216uv^3(3u + v)

We can see that both terms have a common factor of (3u + v), so we can factor it out:

(162u^2v + 216uv^3)(3u + v)

And that is the simplified form of the expression.  

 May 9, 2023

To get your result it would seem to be a whole lot easier to simply remove a factor 3 from the second bracket and to move that into the first bracket.

However the result would then be

\(\displaystyle (162u^{2}v+54uv^{2})(3u+v),\)

which is different from your result, and not much of an improvement on the original.

I think that they are looking for something different.

Notice that 18uv is a common factor of the two terms in the first bracket, remove that and see where that leads.

Tiggsy  May 9, 2023

Hi Tiggsy and George.


Assuming that 3u + v neq 0, simplify (54u^2 v + 18uv^2)(9u + 3v).


In a standard Australian High School I taught that simplify means "Get rid of the brackets and then collect like terms."

I think that is all that is required. 

 May 11, 2023

Hi Melody.

I take your point regarding simplification, and if the numbers, in this question, were different, I might agree with you.

If the thing to be simplified was \((53u^{2}v+18uv^{2})(10u+3v)\)

for example, all that you can do is remove uv from the first bracket or/and get rid of the brackets and collect up like terms. 

What we are given though is \((54u^{2}v+18uv^{2})(9u+3v),\)

and this permits

\(18uv(3u+v).3(3u+v) \\ =54uv(3u+v)^{2}.\)

My feeling is that's what they are looking for.

 May 11, 2023

Hi Tiggsy, 

You are probably correct ,  either way the instructions are poor.

Melody  May 12, 2023

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