6 Questions
What is the primary characteristic of a symmetric equation?
It remains true when the variables are interchanged.
If an equation is symmetric in a and b, then it is also symmetric in b and c.
False
What is the advantage of using symmetric equations in algebra?
They are useful in simplifying algebraic expressions and equations.
A symmetric equation involving monomials is called a ______________ symmetric equation.
monomial
Match the following types of symmetric equations with their examples:
Monomial symmetric equation = a^2 + b^2 = b^2 + a^2 Binomial symmetric equation = (a + b)^2 = (b + a)^2 Polynomial symmetric equation = a^3 + b^3 = b^3 + a^3
Symmetric equations have applications in combinatorics and geometry.
True
Study Notes
Symmetric Equations
Definition
A symmetric equation is an algebraic identity that remains true when the variables are interchanged.
Example
The equation (a + b)^2 = a^2 + 2ab + b^2 is symmetric because it remains true if we interchange a and b.
Properties
- If an equation is symmetric in
aandb, then it is also symmetric inbanda. - If an equation is symmetric in
aandb, anda = b, then the equation reduces to an identity ina.
Types of Symmetric Equations
-
Monomial symmetric equations: Equations involving monomials (expressions with only one term) that are symmetric in the variables. Example:
a^2 + b^2 = b^2 + a^2. -
Binomial symmetric equations: Equations involving binomials (expressions with two terms) that are symmetric in the variables. Example:
(a + b)^2 = (b + a)^2. -
Polynomial symmetric equations: Equations involving polynomials (expressions with multiple terms) that are symmetric in the variables. Example:
a^3 + b^3 = b^3 + a^3.
Importance of Symmetric Equations
- Symmetric equations are useful in simplifying algebraic expressions and equations.
- They can be used to prove algebraic identities.
- They have applications in various areas of mathematics, such as combinatorics, number theory, and geometry.
Symmetric Equations
Definition
- A symmetric equation is an algebraic identity that remains true when the variables are interchanged.
Properties
- If an equation is symmetric in
aandb, then it is also symmetric inbanda. - If an equation is symmetric in
aandb, anda = b, then the equation reduces to an identity ina.
Types of Symmetric Equations
Monomial Symmetric Equations
- Involve monomials (expressions with only one term) that are symmetric in the variables.
- Example:
a^2 + b^2 = b^2 + a^2.
Binomial Symmetric Equations
- Involve binomials (expressions with two terms) that are symmetric in the variables.
- Example:
(a + b)^2 = (b + a)^2.
Polynomial Symmetric Equations
- Involve polynomials (expressions with multiple terms) that are symmetric in the variables.
- Example:
a^3 + b^3 = b^3 + a^3.
Importance of Symmetric Equations
- Useful in simplifying algebraic expressions and equations.
- Can be used to prove algebraic identities.
- Have applications in various areas of mathematics, such as combinatorics, number theory, and geometry.
Learn about symmetric equations, their properties, and examples. A symmetric equation remains true when variables are interchanged.
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