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.