Equations play a crucial role in mathematics as they represent the equality between two expressions, which can be either algebraic or numerical in nature.
The first type, algebraic expressions, involves variables, constants, coefficients, and algebraic operations such as addition, subtraction, multiplication, and division. Take a look at these examples:
On the other hand, numerical expressions consist of only numbers and algebraic operations. Here are some examples:
A polynomial equation is a combination of multiple terms, including variables, positive integer exponents, and coefficients, with only three algebraic operations allowed: addition, subtraction, and multiplication. An equation cannot be classified as a polynomial if it contains a radical, negative exponent, divided variable, or fractional exponent.
The degree of a polynomial equation determines its type, referring to the highest exponent of a variable in the equation. Let's explore the three most common types:
Polynomial equations come in various forms, and those with a degree higher than 3 are simply referred to as a polynomial equation. The standard form for all polynomial equations is:
anxn + an-1xn-1 + ... + a2x2 + a1x + a0 = 0
For instance, an intriguing example is the equation x7 + 2x4 - 3x + 5 = 0, recognized as a "septic" polynomial equation due to its degree of 7.
Linear equations, a specific type of polynomial equation, contain terms with a power of 1. These equations can be expressed in various forms depending on the number of variables involved.
Solving a linear equation involves finding the value of the variable by simplifying both sides of the equation, rearranging terms to combine like terms, and using multiplication or division to isolate the variable. The graph of a linear equation is always a straight line extending infinitely, as shown below.
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