Signs with Division
Algebra Help: Section 2.20
Learn to apply signs with algebraic division. In the previous examples, the dividend and divisor were considered positive by default. With division, apply the law of signs in multiplication.
The following four examples demonstrate the law which governs the sign of the quotient.
1:
+ ab ÷ (+a) = + b similar to + a * (+b) = + ab
2:
- ab ÷ (-a) = + b similar to - a * (-b) = + ab
3:
+ ab ÷ (-a) = - b similar to + a * (+b) = + ab
4:
- ab ÷ (+a) = - b similar to - a * (+b) = - ab
From these examples, we make the following infernces.
First + divided by + gives +.
Second - divided by - gives +.
Like signs produce +.
Third + divided by - gives -.
Fourth - divided by + gives -.
Unlike signs produce -.
These principles may be deduced from the nature of the signs themselves, by taking another view of division.
Division, considered in it's most elementary sense, is not merely the converse of multiplication. Division is a short process of finding how many times one quantity can be subtracted from another quantity of the same kind.
When the subtraction is possible, and decreases the numeric value of the minuend, bringing it nearer to zero, the operation is real, and must be marked positive, with a plus sign.
When the subtraction is not possible, without going farther from zero, or more negative, we must take the converse operation. When applying the converse operation to the subtraction, mark the quotient negative, with a minus sign.
1. For example, divide 18a by 6a.
Determine how many times 6a can be subtracted
from 18a. We can actually subtract 6a
from 18a 3 times. Therefore the result's positive 3.
2. Divide -18a by -6a.
Here again, the subtraction can actually be performed three times.
Therefore the number's positive 3.
3. Divide -18a by +6a.
Here subtraction will not reduce the dividend toward zero.
Subtracting a positive value from a negative value, creates a result which is farther
negative.
Instead subtraction will reduce the dividend toward more negative values.
However addition will reduce the dividend toward zero.
Therefore perform addition three times.
Yet the addition operation is the the converse of the proposed operation.
We want to subtract 6a from -18a.
Therefore mark the quotient with the converse sign, minus.
The result equals -3.
4. Divide 18a by -6a.
Subtracting -6a from 18a,
will not reduce 18a.
The converse operation will reduce 18a.
Therefore subtract +6a,
than assign - to the quotient,
as -3.
The previous examples demonstrate the following rules.
- Divide the coefficient of the dividend by the coefficient of the divisor, for the coefficient of the quotient.
- Include the letters of the dividend in the quotient.
- With similar literals, subtract exponents of divisors from exponents of dividends. If the exponent of a literal becomes zero, then don't include that literal in the quotient.
- If the signs of the terms are alike. The quotient is positive.
- If the signs of the terms are unalike. The quotient is negative.
NOTE - If the dividend does not exactly contain the divisor, the division may be indicated by writing the dividend above a horizontal line, and the divisor below, in the form of a fraction. The result obtained may be simplified by canceling all factors common to the two terms.
4a2b5c ÷ 6a2b4c2 =
2b ÷ 3c
WRITE AS:
4a2b5c 2b _____ = ____ 6a2b4c2 3c
We haven't covered negative exponents yet. However
multiply each literal by its exponent to understand
why c remains in the divisor.
c c 1 ___ = ___ = ___ c2 c*c c
Page 54-55. Division's covered to page 75
Page 54-55. Division's covered to page 75
Help with Algebra Homework
Free tutorials provide math help for math problems. Tutorials start with algebra 1. Algebra tutorials include word problems related to algebra topics, as they're introduced. See the menu, above for algebra math equations, and explanation, by section.
Windows algebra games, designed to help with classwork, coming soon!
Windows games, in progress, work with this set of free tutorials, modified
from the public domain text book titled,
New Elementary Algebra
containing the rudiments of the science for schools and academies.
By Horation N Robinison, LL. D. Ivison, Blakeman & Company, Publishers, New York and Chicago.
