Division: Greatest Common Divisor
Algebra Help: Section 2.29
Learn to find the greatest common divisor with algebra. A Common Divisor of two or more quantities is a quantity that will exactly divide each of them.
The Greatest Common Divisor of two or more quantities is the greatest quantity that will exactly divide each of them.
It is evident that if two or more quantities be divided by their greatest common divisor, the quotients will be prime to each other.
1. What is the greatest common divisor of
4a2b3cd, 48a4b2c3x,
and
12a3b2cd2?
OPERATION:
4a2b3cd = 22 * a2 * b3 * c * d 48a4b2c3x= 3 * 24 * a4 * b2 * c2 * x12a3b2cd2= 3 * 22 * a3 * b2 * c * d2 _____________________________________ 4a3b2c = 22 * a2 * b2 * c
ANALYSIS -
We resolve the quantities into their component factors,
and write all the powers of each factor in the same column. By
inspection we perceive that all the quantitites contain at least the second
power of 2, and we write 22 underneath as a factor of the greatest common
divisor sought. All the quantitites contain at least the second power
of a, and we write a2 underneath. All the quantitites contain at least the
second power of b and the first power of c, and we write these factors
underneath. And since these are all the common factors, their product,
4a3b2c , must be the greatest common divisor of the given quantities.
2. What is the greatest common divisor of 3ac2(x4 - c4),
and a2cx2 - a2c3?
OPERATION:
3ac2(x4 - c4) = 3 * a * c2 * (x2 + c2) * (x + c) * (x - c) a2cx2 - a2c3 = a2 * c * (x + c) * (x - c) ____________________________________________________________________ ac(x2 - c2) = a * c * (x + c) * (x - c)
ANALYSIS -
Resolving the quantities into factors as before, we readily
perceive that the only common factors are a,c,(x + c), and (x - c).
The product of these factors ac(x2 - c2) must therefore be the greatest
common divisor sought.
From these examples we deduce the following.
Rules
- Resolve the given quantities into factors, and write all the powers of each factor in the same column.
- Multiply together the lowest powers of all the common factors, and the product will be the greatest common divisor sought.
NOTE - Define a common divisor. The greatest common divisor. Give analysis of Example 1.
Page 72-73
Page 72-73
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.
