Terms:
- Term: something separated by a + or -
4y3 - 3xy + x - 9
- In the above, there are 4 terms
- 4y3
- -3xy
- x
- -9
-
Terms take the sign in front (to the left) of them
- Constant terms: unchanged terms (no variable)
-
Variable terms: term that can change depending on the number plugged in
- Every variable term has a coefficient and a variable part
| Variable term |
Coefficient |
Variable part |
|---|---|---|
| 4y3 |
4 |
y3 |
| -3xy |
-3 |
xy |
| x |
1 (hidden coefficient) |
x |
| -x |
-1 (hidden coefficient) |
x |
Combining like terms:
- Like terms: terms with the same variable part
- That means the same variable, raised to the same exponent
- Any constant terms are automatically like terms (due to the lack of coefficients
13ab + 4 - 3ab - 10
- In the above, 13ab and -3ab are like terms
- As well as the constants 4 and -10
2xy + 9yx
-
The above are like terms as well, due to the commutative property of multiplication
-
Terms can be combined by adding the coefficients of like terms
- When combining like terms, the variable part does not change
Multiplication
- To multiply a constant by a variable term (or vice versa), multiply by the coefficient
- Keep the variable part, but this gets the correct coefficient.
See example:
5(4x) -> (5 * 4) * x -> 20 * x -> 20x
Distribution
- Distributive property: as long as you multiply the outside term by each of the inside terms, you’ll get the same answer
- Outside factor by each of the inside terms
- This works not only with numbers, but also “in general”
- See here for quick explanation (1:20:40 lc 3.1)
See example
2(3 + 4) -> (2 * 3) + (2 * 4) -> 6 + 8 -> 14
Works just the same “in general” (i.e., for variables?)
a(b + c) -> (a * b) + (a * c)
- To understand why this works, see the associative property of multiplication
-
See this note and this clip
-
Basically, multiplication doesn’t care how numbers are grouped
-
Reminder: when distribution involves variables, multiply the coefficients
When distributing any number (especially negative), it’s important to distribute with the sign
-4(x - 2y) -> (-4 * x) - (-4 * 2y) -> -4x - (-8y) -> -4x + 8y
- There’s a shortcut to get to the same answer without all the intermediate steps:
- Treat what’s in the parentheses as positive/negative rather than plus/minus
- Distribute with that understanding
- Read the answer with plus/minus again
For the previous example
-4(x - 2y)
- That would be:
- “negative 4 times positive x” which outputs: -4x
- “negative 4 times negative 2y” which outputs: +8y
- This gives us the same answer of: -4x + 8y
- Which is now read as “negative 4x plus (positive) 8y”
- The actual correct answer; reached via the above shortcut
- Which is now read as “negative 4x plus (positive) 8y”