Question

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

Answer 1

which aspect is troubling you here?

Answer 2

all of it

Answer 3

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.

Answer 4

no

Answer 5

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 "-"

Answer 6

e.g.:

(1+4-5) + (6-3) - (2+4-1) = 1+4-5 + 6-3 -2-4+1

Answer 7

does that make sense so far?

Answer 8

yes

Answer 9

good, so now try to use these rules to remove the braces in your expression

Answer 10

\[2x ^{4}y ^{3}+4x ^{3}y ^{4}-5x ^{4}y ^{3}+6x ^{3}y ^{4}\]

Answer 11

perfect!

now you just need to combine "like" terms

Answer 12

do you know what "like" terms are?

Answer 13

7x^14y^14

Answer 14

no - that is not quite right

Answer 15

fudge

Answer 16

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 \(x^2y^3\)

Answer 17

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?

Answer 18

no

Answer 19

is it one specific part here that is unclear or the whole thing?

Answer 20

the last explanation

Answer 21

ok - so you understand how I picked out the two unique combinations of powers of x and y in my initial expression - correct?

Answer 22

that is where i am having the issue

Answer 23

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.

Answer 24

yes

Answer 25

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?

Answer 26

ok i think that make sense

Answer 27

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?

Answer 28

yes

Answer 29

so i would come up with -3x^4y^3 +7x^3y^4

Answer 30

good - now we just simplify our new expression by combining all terms involving \(A\) and separately all terms involving \(B\).

this leads to: \(3x^3y^2+2x^2y^3-x^3y^2=3A+2x^2y^3-A=3A+2B-A\)
= \(3A-A+2B\)
= \(2A+2B\)

Answer 31

is t hat right

Answer 32

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?

Answer 33

-3A+10B

Answer 34

correct - now just replace A and B by what they represent

Answer 35

ok that makes sense

Answer 36

the "trick" is to recognise these "unique" combinations of terms - that is what is meant by "like" terms in an expression.

Answer 37

got it right thanks asnaseer

Answer 38

it is similar to saying:

2*apples + 3*pears - 5*apples = -3*apples + 3*pears

i.e. you cannot combine "apples" and "pears"

Answer 39

ok - glad you got it now