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subtract the polynomials (2x^4y^3+4x^3y^4)-(5x^4y^3-6x^3y^4)

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which aspect is troubling you here?
all of it
ok, first step is to remove the braces. do you know how to remove the braces? especially the braces with the minus sign outside of it.

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Other answers:

ok - the rule here is that any expression within braces remains "as-is" when the braces are removed unless the braces have a minus sign outside of them. In that case all the signs inside the braces change - i.e. all "-"'s change to "+" and all "+"'s change to "-"
e.g.: (1+4-5) + (6-3) - (2+4-1) = 1+4-5 + 6-3 -2-4+1
does that make sense so far?
good, so now try to use these rules to remove the braces in your expression
\[2x ^{4}y ^{3}+4x ^{3}y ^{4}-5x ^{4}y ^{3}+6x ^{3}y ^{4}\]
perfect! now you just need to combine "like" terms
do you know what "like" terms are?
no - that is not quite right
the way to think about this might be to imagine each unique combination of products of powers of x and y as a separate entity. e.g. in \(3x^3y^2+2x^2y^3-x^3y^2\) we have two unique combinations, namely: \(x^3y^2\) and \(x^2y^3\)
think of each of these combinations as a new "variable". so, in the example I gave, let \(A=x^3y^2\) and \(B=x^2y^3\). we can then write my expression as: \(3x^3y^2+2x^2y^3-x^3y^2=3A+2B-A=2A+2B\) make sense?
is it one specific part here that is unclear or the whole thing?
the last explanation
ok - so you understand how I picked out the two unique combinations of powers of x and y in my initial expression - correct?
that is where i am having the issue
ok - lets take a look in more detail. we have the expression: \(3x^3y^2+2x^2y^3-x^3y^2\) do you agree that this contains terms that involve multiples of \(x^3y^2\) and \(x^2y^3\) only? i.e. there are no other "combinations" of powers of x and y within my expression.
now, the next step I took was to replace all occurrences of \(x^3y^2\) by a new variable that I called \(A\). this leads to: \(3x^3y^2+2x^2y^3-x^3y^2=3A+2x^2y^3-A\) make sense so far?
ok i think that make sense
good - the next step was to tackel the remaining unique combination which was \(x^2y^3\) - I replaced all occurrences of this with another new variable that I called \(B\). this leads to: \(3x^3y^2+2x^2y^3-x^3y^2=3A+2x^2y^3-A=3A+2B-A\) ok so far?
so i would come up with -3x^4y^3 +7x^3y^4
good - now we just simplify our new expression by combining all terms involving \(A\) and separately all terms involving \(B\). this leads to: \(3x^3y^2+2x^2y^3-x^3y^2=3A+2x^2y^3-A=3A+2B-A\) = \(3A-A+2B\) = \(2A+2B\)
is t hat right
your expression after removing the braces was:\[2x ^{4}y ^{3}+4x ^{3}y ^{4}-5x ^{4}y ^{3}+6x ^{3}y ^{4}\] so if you first replace \(x^4y^3\) by \(A\) and \(x^3y^4\) by \(B\) then you would get:\[2x ^{4}y ^{3}+4x ^{3}y ^{4}-5x ^{4}y ^{3}+6x ^{3}y ^{4}=2A+4B-5A+6B\] what do you think that simplifies to in terms of A and B?
correct - now just replace A and B by what they represent
ok that makes sense
the "trick" is to recognise these "unique" combinations of terms - that is what is meant by "like" terms in an expression.
got it right thanks asnaseer
it is similar to saying: 2*apples + 3*pears - 5*apples = -3*apples + 3*pears i.e. you cannot combine "apples" and "pears"
ok - glad you got it now

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