Algebraic Fractions
Signs, Exponents, Radicals and
Complex Numbers
Algebraic Fractions
• Let us work some fractions out using
– Factoring
– Cancelling
– Reducing
These are some methods of simplification.
Simplifying Fractions
Factoring By Using Identities
More Identities
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A2 + B2 + 2AB = (A + B)2
A2 + B2 – 2AB = (A – B)2
These are also written as:
A2 + 2AB + B2 = (A + B)2
A2 – 2AB + B2 = (A – B)2
We are more familiar with
x2 – 2xy + y2 or x2 + 2xy + y2
Examples
Multiplication of Polynomials
The concept discussed in the prior slide comes
from the fact that we can manipulate
polynomials and binomials like the (A + 2)2
(x + y)(a + b + c) would require us to multiply out
each of the variables. That is x to a, x to b and x
to c then y to a, y to b and y to c.
ax + ab + ac + ay +by + cy. The (A + 2)(A + 2) is A2
+ 2A + 2A + 22 adding like terms which are the
2A + 2A is 4A.
More Complex Example
Signs
Proportions
Powers
Powers are important as well. Our examples
have shown variables with powers.
• A2 + B2 + 2AB = (A + B)2
• A2 + B2 – 2AB = (A – B)2
The equations all have powers. The 2 in A2
means squared or multiplied twice such as A*A
If it was a 3 it would be A*A*A or cubed.
Fractions also can have powers.
Powers and Radicals
Radicals and Complex Numbers
Powers of j
j1 = j
j2 = -1
j3 = -j
j4 = 1
The cycle repeats:
j5 = j
j6 = -1
j7 = -j
j8 = 1
Phasors and Vectors
Both complex numbers and their graphical
representations (a directed line between origin
and complex point) are called phasors in
electronics.
Vectors are not phasors since vectors are
plotted on cartesian coordinates to represent
real numbers.
Adding phasors: (2 + j3) + (6 + j4) = 8 + j7.
Remember real + real and imaginary + imaginary
Thank you!
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