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Algebra for Adults:
Part 8, Factors, terms, coefficients, and exponents:
January 20, 2003:
First the practice expressions from Part 7.
Evaluate the following:
3(4 - 2 + 8) + 2(a - c)/c where a = 5 and c = 2
3(10) + 2(5 - 2)/2 = 3(10) + 2(3)/2 = 30 + 3 = 33
6(a + b/2) - 2(b - c)/c where a = 6, b = 8, and c = 3
6(6 + 8/2) - 2(8 - 3) = 6(6 + 4) - 2(5) = 60 - 10 = 50
a(2b + 4c)/(a - 2) where a = 4, b = 2, and c = 3
4(2(2) + 4(3))/(4 - 2) = 4(4 + 12)/2 = 64/2 = 32
x(37y - 22z)/(x - 4) where x = 4, y = 5, and z = 257
This last one is just a little curve ball to help you remember
an old rule.
The rule is:
Division by zero is undefined or impossible.
Never mind the rest of the problem. As soon as we substitute
the values for the letters we can see that the divisor is 4 -
4 = 0.
Thus, the answer is undefined.
Now, we can look at a few definitions that are used in algebra.
First we define a term:
A term is a number or the product of numbers.
So 6, 17y, xy, and 7bc are all terms.
Well sure. This is not much different from our definition of a product in basic math. The difference is, a term can be a single number or a product of numbers.
Next we define a factor of a term:
A factor of a term is any of the numbers multiplied to form the
term.
So, 7 and b and c are all factors of the term 7bc.
Did we already know this?
Sure. This agrees with our definition of factor from basic math.
Factor:
One of two or more numbers that are multiplied to produce a product.
Let's notice one thing about this. We can have complex factors.
For example:
In cd(a + c), (a + c) is a factor.
The factors of this expression are c, d, and (a + c).
We can also have more than one term in an expression.
For example:
In the expression c + cd - (c - d) the terms are c, cd, and (c
- d).
Now we can define coefficient:
Any factor or group of factors of a term is a coefficient of the
product of the remaining terms.
Holy Mackerel!
It is not really as complicated as that formal statement makes
it seem.
Lets take some examples to clear it up.
In the expression 4(x + y) 4 is a coefficient of (x + y) and (x + y) is a coefficient of 4.
In the expression 5cd, 5 is a coefficient of cd and cd is a coefficient of 5.
Now, we can have literal coefficients and numerical coefficients.
In the above example, can you guess that 5 is the numeric coefficient
of cd?
Sure!
Then cd must be the literal coefficient of 5.
Makes sense.
So far, so good!
Now lets's take a look at exponents.
In basic math, we learned to raise a number by a power or multiply
it by itself a number of times. We indicate that operation with
a superscript number.
For example:
a8pr1
When we multiplied a number by itself we called it squaring
the number.
Likewise we learned that we could raise a number to any power
we wished.
For example:
a8pr2
Now we can formalize that a bit.
We call the superscript number the exponent.
We call the number being multiplied the base.
The result of the operation is called a power of the base.
For example:
125 is the third power of 5.
5 is the base.
3 is the exponent.
Finally, in certain case we can combine terms.
We can do this if the terms have identical literal factors and
exponents.
For example:
We can combine 6 + 7 + 2 into a single term 15.
Of course, we also call that addition and it is obvious.
It is not quite as obvious when we add literal factors to the
mix.
For example:
We can combine 12x + 7x into a single term 19x.
What we did was add the numeric coefficients.
We could do that because the literal coefficients were identical.
We can prove this is true by substituting any number for x and
solving the expression.
For example:
Where x = 3 the expression becomes 12(3) + 7(3) = 19(3) or 36 + 21 = 57 or 57 = 57.
It should be clear that we cannot combine terms with unlike
coefficients or exponents.
For example:
We cannot combine 12x + 7y.
Now, let's quit before we get snowed under.
Here are a few teasers for you.
What are the terms of these expressions?
7 + 3x - 2(4 + 3)
45x + 3(2 - 4) - 15
64 + 23 + 18 - 22
What are the factors of the following expressions.
a(c + 4)
9ab
2(x - c)(y + 13)
What are the base, exponent, and power of the following.
a8pr3
a8pr4
a8pr5
Combine the following terms:
27 z+ 3 z
a8pr6
2(a + b) + 6(a + b)
Back to Algebra for Adults.
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