Introduction of Mathematical Expressions
A mathematical expression is the combination of a variable and number in a logical way using basic mathematical operators such as addition, multiplication, subtraction and division. Each and every such expression will have a solution, which is the value of the variable that satisfies the given expression. Mathematical expressions are in the form of 5x + 8 or x/2-3, 7x-4, etc., These can also be called polynomials. There are different types of polynomials. We can categorise them based on the number of terms and the highest degree of the variable.
If there is only one term in an expression, then it is called a monomial, if the number of terms in an expression are two, then it is called binomial and if the number of terms is equal to three, then it is called a trinomial and so on. Apart from these, we can also classify them based on the degree of the variable. If the highest degree of expression is one, then it is called a linear polynomial, if the highest degree is two, then it is called a quadratic polynomial and if the highest degree is three, then it is called a cubic polynomial.
We can find solutions for all those types of polynomials using different methods. One of the frequently used methods is factorisation. By factoring polynomials we can get the values of a variable, which are often called solutions of that polynomial. But this method is not applicable to linear polynomials since we can get the solution for these types of polynomials just by equating to zero. Factorisation method also results in linear equations at the end. This method is very popular in maths as this will be used in many concepts. For example, while solving equations, trigonometric equations and so on.
There are mainly four operations that can be performed over polynomials, they are addition, subtraction, multiplication and division. Addition or subtraction of polynomials involves adding or subtracting the like terms, i.e. the terms with the same variable and degree of two or more given polynomials, respectively. In both the operations, we can get the polynomial of the same degree in the result unless there are no equal terms while subtracting. Otherwise, the highest degree of the polynomial will not change.
Multiplication of two or more polynomials always results in an increase in the degree of polynomial unless there is a constant polynomial. While performing this operation we will use the distributive property of multiplication under addition or subtraction. There are certain conditions for dividing polynomials. The division of polynomials may or may not give polynomial as a result.
This concept can be applied in many real-life situations and also in solving word problems in mathematics. There are certain exercises in Maths where we need to translate the words into mathematical expressions. There are different types of word problems that can be translated into mathematical expressions with different operators such as plus (for sum), minus (for difference), and so on. For example, 8 times the sum of the number x and 7, the difference of 5 times of x from 14. The respective expressions of these word phrases are 8x + 7 and 14 – 5x.
In modern mathematics, particularly in computer algebra, formulas are considered and presented as an expression that can be estimated to be true or false, based on the numerical values that are given to the variables involved in the expressions. Coming to the variables in the expression, these can be classified as free variables and bound variables. For free variables, a combination of numerical values are given based on which the expression can be evaluated. However, for some combination of numerical values of free variables, the expression may not be evaluated exactly that means undefined.
This concept is also used in summarising certain types of statements which involve some reasoning. Following are some more example concepts of word phrases in maths, wherever needed, to form mathematical expressions.
- Age problems: Finding how old are the persons as of now or a few years back or after some years
- Perimeter, Area, and Volume: Expression with basic geometric formulas
- Distance problems: Expressing the relationship between time, speed and distance using a variable and so on.