Algebraic Equations
Algebraic Equations
Today we are going to learn about Algebraic Equations. Algebra is a very important part of
mathematics. Equations are used very frequently in mathematics. Equations are the basic and
main tool of algebra.
Equations are can be balanced or unbalanced. To balance an equation we need to use
variables. A variable is any unknown quantity which are represented by using alphabets such
a, b, c etc.
An algebraic equation is also known as polynomial equations. Polynomial equations are
basical y comes in a form P = R; in this equation P and R are the polynomials over this field.
These P, R polynomials can be multivariate.
For instance we have x^4 + x*y/4 = y^2/2 - x^2y is an algebraic equation.
Similarly px^3 + qx + r = 0 is also an algebraic equation.
For the equality of two expressions we use a set of variables and a set of algebraic
operations. We apply these algebraic operations on set of variables.
Know More About :- Divergence Theorem


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The algebraic operations are basically addition, multiplication, subtraction, division, root
extraction and rising to a power of a number or variable. For example: y^2 - 1, (x^3 y^4 + 2
x*y) / (y - 2) = 14.
Polynomial equations are the special case of algebraic equations. The polynomial equations
have a form ax^n + bx^n-1 + ....... +hx + I = l.
Such equations have so many solutions, it means if these have n powers or n degree then
they have n numbers of solutions.
To solve algebraic equations we should use so many variables or set of variables so that we
can find out a variable or set of variables which if substituted in the equation; reduces it to an
identity.
Two equations are said to be equivalent if they have the same solution or same set of
solutions. For instance we can say that P = R is equivalent to P - R = 0.
We can convert an algebraic equation over the rationales to an equivalent equation. Equation
wil be consisting of the coefficients which are basically integers.
For instance: We have an equation: x^4 + x*y/2 = y^3/3 - yx^2 + x^2 - 1/7
The above equation is an equation over rational this can be converted to equivalent one. For
this we need to multiply this equation through by 2*3*7 = 42.
By multiplying that equation by 42 we wil get: 42 x^2 + 21 x*y - 14 x^3 + 42 y^2 + 6 = 0
The algebraic equations are different from other ordinary equations where the solution is the
values of variables of that equation for which the equation becomes true. In algebraic
equations solution P = 0 are basical y the roots of the polynomial P.
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Whenever we solve such equations we have to give the specification of the set of variables in
which the actual solutions are allowed to present.
For instance An algebraic equation over rationals; for this we can look for the solution that
simply may consisting of the variables and where variables are integers.
Also we may also find a solution where variables are complex numbers. The algebraic
equation over the rational consisting of only a single variable is also known as the uni variant
equation.
So in today's session we learnt about the algebraic equations;


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