Hey it’s Professor Dave, let’s learn some algebraic properties.
Remember when we learned about a few different mathematical properties and how they pertained
to numbers?
Let’s learn how they will be important in algebra when manipulating variables.
The most important of these properties will be the distributive property.
This told us how a number could be distributed across a parenthetical sum or difference.
This didn’t matter too much for arithmetic, because four times the quantity of two plus
three is certainly equal to four times two plus four times three, but there was nothing
stopping us from adding two and three first, and then multiplying by four.
We should get twenty no matter which method we choose.
But with algebra, there are variables that must remain as they are, and can’t be combined
with numbers, so if we have four times the quantity of two X plus three, the only other
meaningful way to express this is by distributing the four across the sum.
That will give us four times two X plus four times three, which will simplify to eight
X plus twelve.
In algebra, we will want to be able to use the distributive property this way, and we
will even want to be able to do it in reverse, by removing some common factor from a sum
or difference.
For example, if we have three X squared plus six X, another way to express this would involve
identifying the greatest common factor of these terms, and factoring it out of the expression.
In this case, we can rewrite these as three times X times X and two times three times
X. Taking everything we find in both terms, the greatest common factor would be three
X, and if we pull a three X out of both terms, meaning we divide each term by three X, we
end up with the three X out here, and then X plus two in parentheses.
We can verify that this worked as expected by then distributing the three X across the
sum.
Three X times X is three X squared, and three X times two is six X.
So we can use the distributive property in a variety of ways to generate equivalent expressions.
Other properties that also apply include the commutative property for addition and multiplication.
Two plus three is the same as three plus two, and two times three is the same as three times
two.
If these become algebraic terms, the commutative property applies in precisely the same way;
the order in which we add or multiply algebraic terms is irrelevant.
Two X plus three is the same as three plus two X, and two X times three is the same as
three times two X.
However, let’s recall that the commutative property does not apply to subtraction or
division, and that will be the case in algebra as well.
We also learned about the associative property, and that will apply in algebra too, which
we will find out later when we have to manipulate equations with lots of terms in them.
Changing the way these are grouped will not matter if we are doing addition or multiplication.
To be thorough, let’s also mention some pretty intuitive properties like the additive
identity property.
This says that you can add or subtract zero to any number or algebraic term, and it will
retain its identity.
Five plus zero is five.
Three X minus zero is three X. Seems obvious, but it will come in handy.
The multiplicative identity property works the same way except with the number one instead
of zero.
Any number or algebraic term times one will give you the same term again.
Four X times one is four X.
And lastly, the inverse property of addition says that anything plus its additive inverse
equals zero, so X plus negative X equals zero, and the inverse property of multiplication
says that anything times its multiplicative inverse equals one, so X times one over X
equals one.
That’s pretty much all we need to know in terms of number properties for algebra, so
let’s get to some equations.