Extended Algebra Topics
We have been combining monomials withmonomials
◦Even when we did 3x + 7 + 21x – 11, we wereactually only combining monomials because 3x and21x are both monomials (we put them together toget 24x) and +7 and – 11 are both monomials (weget – 4 when we put them together).
◦So, even though our answer (24x – 4) is a binomial,we really only combined monomials.
◦Today, we will look at binomials being added orsubtracted to other polynomials.
3x + 9x
7y + 11y
9n – 3n
123p – 65p
We always did either the first term plus thesecond or the first term minus thesecond…easy right?
3x + (2x + 7)
Continue to do the first part plus the secondpart…in other words, 3x plus all of 2x + 7
Combine what can be combined…the 3x andthe 2x. Thus, do 3x + 2x to get 5x.
Since the 7 has no one to “team up” with, youjust tack him on the end.
Answer: 5x + 7
12y + (3y + 90)
89m + (-3m + 11)
-78y + (-9y – 123)
99r + (7p – 23)
45h + (15h – 60)
BE CAREFUL WITH SUBTRACTION
15m – (13m + 7)
The parenthesis indicate that you are subtractingthe whole 13m + 7 from the 15m.
Thus, you have to do 15m – 13m to get 2m
Since there is no term like 7 in the first part ofthe problem, you must treat it as 0 – 7, whichgets you -7 instead of +7
It looks like this….15m + 0 – (13m + 7)
So, 15m – 13m is 2m…+ 0 – 7 is -7
Answer is 2m - 7
11m – (3m + 8)
32x – (12x + 9)
45p – (11p – 10)
-32k – (5 + 23k)
88m – (23m – 56)
(5m + 9) + (3m + 10)
With this situation, the first part has abinomial as does the second part. Thus, youare just combining things that are like
5m + 3m and +9 + 10
8m +19
Answer is 8m + 19
Watch your signs…remember the role of theminus sign can change into a negative sign.
(5m + 6) + (3m + 65)
(11x – 5) + (9x + 13)
12x + 19 + (-20 – 19x)
(-43p – 56) + (124p + 14)
(-67n – 88) + (-12n – 64)
(5m + 9) – (2m + 6)
Remember, you have all of the first part,minus all of the second part.
Thus, start with the 5m and subtract the partof the second binomial that can be subtractedfrom 5m (aka the 2m)….5m – 2m
Then, do the positive 9 – the positive 6or…9 – 6
Answer is 3m + 3
WATCH OUT FOR THE SIGNS!
(6p + 7) – (2p + 4)
(13x + 9) – (12x + 6)
(15m – 13) – (11m + 4)
(-34r – 20) – (-21r + 30)
(45p + 43) – (45p – 36)
If an example says in words….”What is 5xsubtracted from 13x?”, how would you dothat?
13x – 5x
Remember, subtracted from was “a star”,which meant that the numbers need to beflipped around…just like less than.
So, 11x + 9 subtracted from 15x + 12 shouldbe written how?
(15x + 12) – (11x + 9)
What is 5x + 13 subtracted from 8x + 11?
What is -4x – 9 subtracted from 11x + 5?
What is 15x – 21 subtracted from 12x – 8?
Remember what we did when we subtractedintegers? We did “add the opposite”. In otherwords, we said that if you change the subtractionsign to addition, and then change the sign on thesecond integer to it’s opposite, we can proceedwith addition.
Let’s apply this to this topic
Example: 32x – (12x + 7)
Changes to 32x + (-12x – 7)
In this case, you change the minus to plus, andyou change EVERYTHING in the second part ofthe problem to it’s opposite. See if you get thesame answer as the previous way.
(5x + 11) – (2x – 9)
(19y – 23) – (9y + 17)
82m – 9 subtracted from 42m + 99
-34w – 19 subtracted from -17w – 89
Remember that you need a commondenominator when you add or subtractfractions, but in this case, you will only needit for the common terms. All other rules stillapply (subtracted from etc)
