Algebraic Equations

Two algebraic expressions are linked using the equal to (=) symbol to form an algebraic equation. Because both sides of the equal sign contain polynomials, an algebraic equation is also known as a polynomial equation. Variables, coefficients, and constants, as well as algebraic operations like addition, subtraction, multiplication, division, and exponentiation, make up an algebraic equation.

The roots or solutions of an algebraic equation are a number or a set of integers that fulfil the equation. We’ll learn more about algebraic equations in this post, including their types, examples, and how to solve them.

Algebraic Equations Formulas:

Some formulas and identities are used to simplify the algebraic equations. These can help speed up the process of fixing a problem. The following are some key algebraic formulas:

(x + y)² = x² + 2xy + y²

(x – y)² = x² – 2xy + y²

(x + y)(x – y) = x² – y²

(x + a)(x + b) = x² + x(a + b) + ab

(x + y)³ = x³ + 3x²y + 3xy² + y³

(x – y)³ = x³ – 3x²y + 3xy² – y³

X³ + y³ = (x + y)(x² – xy+ y²)

(x + y + z)² = x² + y² + z² + 2xy+ 2yz + 2zx

Discriminant: b² – 4ac

Algebraic Equations Examples:

X² – 9x = 5 is a univariate algebraic equation while y²x – 5z = 3x is an example of a multivariate algebraic equation.

Types of Algebraic Equations:

Based on the degree of the equation, algebraic equations can be categorised into distinct categories. The highest exponent of a variable in an algebraic equation is known as the degree. If the degree is determined by the equation x⁴ + y³ = 35, the degree will be 4. The exponent of the constant or coefficient is ignored while determining the degree. The degree of an algebraic equation determines how many roots it has. There will be 5 roots in an algebraic equation with a degree of 5.

There are 5 Basic Rules for Algebraic Equations Mentioned below:

Commutative Rule of Addition

Commutative Rule of Multiplication

Associative Rule of Addition

Associative Rule of Multiplication

Distributive Rule of Multiplication

Algebraic Equations  of Various Types  are Mentioned below:

1.Monomial: when there is only one term in algebraic expression.

Example: (9x, 8xy), etc., are monomials.

2. Binomial: when there are two terms in algebraic expression.

Example: ((2a+3b),(8 – 3x),({x²} – 4x{y²})), etc., are binomials.

3. Trinomial: when there are three terms in algebraic expression.

Example: ((a, + 2b + 5c), (x + 2y – 3z), ({x³} – {y³} – {z³})) etc., are trinomials.

4. Quadrinomial: when there are four terms in algebraic expression.

Example: ((x + y + z – 5), ({x³} + {y³} + {z³} + 3xyz)) etc., are quadrinomials.

5. Polynomials: when there are two or more terms  in algebraic expression is known as a polynomial. 

Polynomial equations have a polynomial expression on each side of the equal sign in them. Variable terms and whole number exponents are found in polynomials. Each polynomial equation can be classified based on the number of terms in the expression:

A monomial is of one term.

A binomial has two terms.

A trinomial has three terms.

Exponential equations are those with an exponential expression on both sides of the equal sign. Exponential equations are polynomial equations with a variable term in the exponents. When the independent variable has a positive coefficient, exponential functions exhibit exponential growth; when the independent variable has a negative coefficient, exponential functions show exponential decline.

A Rational expression appears on each side of the equal sign in rational equations. When algebraic equations assume the form b(x) / d(x), where b(x) and d(x) are both polynomials, they are rational. Asymptotes are points on a graph of rational equations where the x and y values approach but never reach.

A Trigonometric expression appears on each side of the equal sign in trigonometric equations. In trigonometric equations, the trigonometric functions tan, cos, sin,cot, cosec, and sec express the ratio between two sides of a right triangle. The ratio is the dependent variable or output, while the angle measure is the independent variable or input. Trigonometric functions are special in that they are periodic, which implies that their graph repeats after a fixed amount of time.

Difference between Algebraic Equation and Algebraic Formula:

Conclusion:

An algebraic equation is one in which two algebraic expressions are connected by the equal sign. Algebra equations are polynomial equations. Univariate and multivariate algebraic equations exist.

Based on the degree, algebra equations are classed as linear, quadratic, cubic, and higher-order equations.