Algebra - Algebraic Expressions

Algebra is a branch of mathematics that generalizes the rules of arithmetic calculations through the use of variables.

A term can assume many forms.

Algebra has its own writing conventions.

Writing different elements together, one after the other, means that they are multiplied. For instance, |6c| means |6\times c| and |7a^2b| means |7\times a\times a \times b.|

Similarly, when a term has no visible coefficient, its coefficient is 1. In other words, |1b = 1\times b = b| since the number |1| is the neutral element of multiplication.

The same applies to exponents. When a variable has no visible exponent, the exponent is simply 1. So, |5x=5x^1| and |10xy^2=10x^1y^2.|

Furthermore, algebraic expressions can be composed of single or multiple terms.

• If there is only one term, the algebraic expression is a monomial.
  Examples: |x,| |8y,| |-4ab,| |\dfrac{9a^2b}{5},| |5a\times 3b,| |8a \div 3|

• If there are several terms, the algebraic expression is a polynomial.
  Examples: |x-y,| |x^3+x^2+x+1,| |25x^2y^4-16,| |15ab + 3c \times 6,| |\dfrac{5a+3b}{2}|

It might appear that the algebraic expression |15ab + 3c \times 6| is composed of 3 terms, but in fact it only has 2. This is because distinct terms must be separated by |+| and |-| symbols. In other words, |3c \times 6| counts as just one term.

Similarly, the expression |8a \div 3| is indeed a monomial. It's best to rewrite this expression using fractional notation as follows:||8a\div 3=\dfrac{8a}{3} =\dfrac{8}{3}a||The coefficient of the monomial is |\dfrac{8}{3}.|

From the statement “Your brother is |2| years older than you,” the following table of values can be constructed:

Note that calculating your brother's age always follows this pattern:||\text{Your age}+2=\text{Your brother's age}||Therefore, if you replace “Your age” with the variable |a,| the algebraic expression representing your brother's age becomes |a+2.| ||a+2=\text{Your brother's age}||The 1^st term is composed solely of the variable |a| and the 2^nd term is a constant term, because your brother is always 2 years older than you, even if your age varies over the years.

• Variable : |x|

• Constant term: |23|

• Algebraic expressions: |4x-5| and |2x+7|

• Equations that express the equality between an algebraic expression and a constant term: |4x-5=23| and |23=2x+7|

• Equations that express the equality between 2 algebraic expressions: |4x-5=2x+7|

In an algebraic expression, terms are considered to be like terms when they are composed of the same variables and assigned the same exponents.

1. If |x = 2| in the algebraic expression |2x +3,| we replace the variable with this value.||\begin{align} 2\boldsymbol{\color{#3b87cd}x}+3 &=2(\boldsymbol{\color{#3b87cd}2})+3 \\ &= 4+3\\ &= 7 \end{align}||Therefore, when |x = 2,| the value of the algebraic expression is |7.|
    

2. If |a=-1.5| and |b=10| in the algebraic expression |3a-\dfrac{b}{5}+2,| 2 substitutions must be made. ||\begin{align}3\boldsymbol{\color{#3b87cd}a}-\dfrac{\boldsymbol{\color{#3a9a38}b}}{5}+2 &=3(\boldsymbol{\color{#3b87cd}{-1.5}})-\dfrac{\boldsymbol{\color{#3a9a38}{10}}}{5}+2\\[3pt] &= -4.5-2+2\\[3pt] &=-4.5\end{align}||Therefore, in this case, the value of the algebraic expression is |-4.5.|
    

3. If |w=5| in the expression |-w^2+w+20,| the |w| must be replaced by |5| wherever |w| appears.||\begin{align}-\boldsymbol{\color{#3b87cd}w}^2 + \boldsymbol{\color{#3b87cd}w}+20 &=-\boldsymbol{\color{#3b87cd}{5}}^2 + \boldsymbol{\color{#3b87cd}{5}}+20\\ &= -25+5+20\\ &=0\end{align}||

1. The coefficient of a term is always written in front of the variables. It's important to note, however, that a coefficient of |1| is not written in an expression.

2. If a term has several variables, it is best to place these variables in alphabetical order.

3. In a polynomial, the terms are arranged in descending order of their respective degrees. If 2 terms have the same degree, they are written in alphabetical order.