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Like Terms
"Like terms" are terms whose variables and exponents are the same.
In other words, terms that are "like" each other.
Note: the coefficients (the numbers you multiply by, such as "5" in 5x) can be
different.
Example:
7x
x
-2x
Are all like terms because the variables are all x
Example:
(1/3)xy2
-2xy2
6xy2
Are all like terms because the variables are all xy2
Unlike Terms
If they are not like terms, they are called "Unlike Terms":
Unlike Terms
-3xy
-3y
12y2
Why they are "Unlike Terms"
← these are all unlike terms
(xy, y and y2 are all different)
Combining Like Terms
You can add or subtract like terms together to make one term:
Example: 7x + x
They are both like terms, so you can just add them by adding the coefficients,
but do NOT add or multiply the variables:
7x + x = 8x
How should you handle expressions or equations with lots of terms like:
3x2 - 7 + 4x3 - x2 + 2
The 1st thing to do is to re-arrange the expression by either putting like terms
next to each other or grouping them together with parenthesis.
Make sure you correctly re-write your negatives or subtraction and your
positives or addition.
So combining or regrouping the expression
3x2 - 7 + 4x3 - x2 + 2 results in
(3x2 - x2) + (4x3) + (2 - 7)
Next, add or subtract like terms:
2x2 + 4x3 – 5
Multiplying Like Terms
We somewhat covered this with exponents.
The key is to make sure that you use the correct rules of multiplying
exponents and that you multiply all coefficients by all coefficients. What
do I mean by that?
Let's forget exponents for a moment.
Say you have 3y × 2y. This is much different from 3y + 2y. Why?
In the addition example, we:
Add the coefficients and we do NOT multiply the variables = 5y
With multiplication, we must multiply both the variables and the
coefficients. In long form:
3y × 2y = 3 × y × 2 × y
If you rearrange this, it becomes: 3×2×y×y
Now multiply similar terms you get: 6y2
Reminder:
Rules of multiplying exponents:
am×an = am+n
(am)n = a(mn)
Dividing Like Terms
The key is to make sure that you use the correct rules of dividing
exponents and that you divide all coefficients by all coefficients. What do I
mean by that?
Let's forget exponents for a moment.
Say you have 4y ÷ 2y.
We must divide both the variables and the coefficients. In long form:
4y ÷ 2y =
4𝑦
2𝑦
Essentially, we REDUCE or combine like terms. Our goal, is to get as many
"1's" as possible.
Here, the 4 is reduced by the 2 and the y's cancel each other out so we are
left with just:
2
Reminder:
Rules of dividing exponents:
am / an = am-n
Simplify the following:
2y3 + x2
y2 + x
Your 1st inkling is going to be to reduce the numerator and the
denominator. Since there is addition and subtraction – DON'T
Let's plug in #'s to show this. Make y = 2 and x = 3
2(23) + 32
22 + 3
= 2(8) + 9
4+3
Can you reduce anything in the num and denom?
1st – let's calculate the real answer which is 25/7.
Now let's see what happens if we reduce the 9 and the 3. So now make it:
16+3 = 19
4+1
5. This is clearly different from 25/7.
What if we multiplied one of the terms in the numerator by 4 and one in
the denominator by 2, could we reduce the 4 and the 2?
2(23) + 32(4) This becomes: 2(8) + 9(4) = 16+36 = 52 = 26
22 + 3(2)
4+6
10
10 5
If we reduced the 4 and the 2, we would have:
2(23) + 32(2) This becomes: 2(8) + 9(2) = 16+18 = 34
22 + 3(1)
4+3
7
7
You can only reduce an expression if it is multiplied throughout the entire
numerator and the entire denominator:
4[2(23) + 32] This becomes: 4[2(8) + 9] = 4[16+9] = 4[25] = 100 = 50
2[22 + 3]
2[4+3]
2[7]
14
14
7
If we reduced the 4 and the 2, we would have a 2 in the numerator and a 1
in the denominator and you would get:
50/7
which not by coincidence, is twice the value of our original expression of
2(23) + 32 = 25/7
22 + 3
Remember – if you have a HUGE division expression or fraction with lots
of terms, you can only reduce it if the numerator and denominator have a
term that is multiplied across every term in the numerator and
denominator.