Algebraic expressions
What is an algebraic expression?
An algebraic expression is an expression which is built from variables, constants and the algebraic operations.
Difference between algebraic expression and algebraic equation
An algebraic expression is a mathematical phase that can contain numbers, variables and not an equal sing .They can´t be solved but they can be simplified. Meanwhile and algebraic equation can be solved and is formed by some algebraic expressions separated by an equal sign.
Types of algebraic expressions
There are many types of algebraic expressions but the main types are:
Based on the nature of the terms:
1. Numeric expression
A numeric expression is an expression that consists in only numbers and operations. Variables are never included.
$\displaystyle 2+4$
$\displaystyle 6\div 3-2$
$\displaystyle 125-3\times 4$
$\displaystyle 2+\left( {5-2} \right)$
2. Variable expression
This is an expression that contains also variables expect numbers.
$\displaystyle 3xz+2$
$\displaystyle z+y$
$\displaystyle 4{{x}^{2}}+2x+1$
$ \displaystyle x+3$
Based on the number of terms:
1. Monomial expressions
This is a type of algebraic expression having only one term.
$\displaystyle 4x$, $\displaystyle 2y$
$\displaystyle 6{{x}^{2}}$, $\displaystyle -3xy$
$\displaystyle 7x{{y}^{3}}z$
2. Binomial expressions
Analgebraic expression having two unlike terms.
$\displaystyle 2x+3$, $\displaystyle 3x+6y$
$\displaystyle xyz+2$, $\displaystyle 4{{x}^{3}}+y$
3. Polynomial expressions
This is an algebraic expression with more than one term and with non-zero exponents of variables.
$\displaystyle 2{{x}^{2}}+3x-2$
$\displaystyle ax+by+c$
$\displaystyle 5x+3xyz+6{{y}^{6}}+3$
Like and unlike terms
In algebra like terms are terms that have the same variables and power.Unlike terms are two or more terms that don´t have the same variables and powers.
Combining like terms
To combine like terms means to add or subtract the terms of the same degree. This means that all terms like x can be combined with other x terms,or all terms y2 can be combined with other y2 terms and all numbers that are not attached to variables can be added or combined as well.
Example 1
$\displaystyle 2x+3-{{y}^{2}}-x+1=$
$\displaystyle (2x-x)+(3+1)-{{y}^{2}}=$
$\displaystyle x+4-{{y}^{2}}$
Example 2
$\displaystyle y+4+{{x}^{2}}+x+{{y}^{2}}+5y-3{{x}^{2}}=$
$\displaystyle ({{x}^{2}}-3{{x}^{2}})+(y+5y)+4+x+{{y}^{2}}=$
$\displaystyle -2{{x}^{2}}+6y+4+x+{{y}^{2}}$
Factors of a term
A term can be a variable or a constant multiplied by a variable or variables. Meanwhile a factor is something which is multiplied by something else.
If 3x is a term then 3 and x are factors of this term.
If 4xyz is a term then 4, x, y, z are factors of this term.
Simplifying an algebraic expression
Expressions are put into their simplest form on purpose to not be too complex. One way of simplifying expressions is to combine like terms by adding and subtracting like we did above. In this way our expression can be in an easier form.
Steps of simplifying an algebraic expression
1. Remove parentheses by multiplying factors.
2. Use exponents rules to remove parentheses in terms with exponents.
3. Combine like terms by adding coefficients.
4. Combine the constants.
Example 3: Simplify the expression $\displaystyle 3(x+1)+{{({{x}^{2}})}^{{^{3}}}}-2\left[ {3({{x}^{6}}+1)} \right]=$ .
Solution: Firstly we need to remove the parentheses by using the distributive property, by multiplying the factors times the terms inside the parenthesis.
$\displaystyle 3x+3+{{({{x}^{2}})}^{{^{3}}}}-2\left[ {3{{x}^{6}}+3} \right]=$
$\displaystyle 3x+3+{{({{x}^{2}})}^{{^{3}}}}-6{{x}^{6}}+6=$
Secondly we use the exponent rule by multiplying the exponents.
$\displaystyle 3x+3+{{x}^{6}}-6{{x}^{6}}+6=$
The next step is to look for like terms and combine them. The terms $\displaystyle {{x}^{6}}$ and $\displaystyle -6{{x}^{6}}$ are like terms because they have the same variable raised to the same power.
$\displaystyle 3x+3+({{x}^{6}}-6{{x}^{6}})+6=$
$\displaystyle 3x+3-5{{x}^{6}}+6=$
Finally we look for any constants that we can combine. We have two constants 3 and 6 that we can combine by adding.
$\displaystyle 3x-5{{x}^{6}}+(3+6)=$
$\displaystyle 3x-5x+9$
Example 4: $\displaystyle {{y}^{4}}+2(x+y)+3y+2{{\left( {{{y}^{2}}} \right)}^{2}}$
Solution
1. $\displaystyle {{y}^{4}}+2x+2y+3y+2{{\left( {{{y}^{2}}} \right)}^{2}}$
2. $ \displaystyle {{y}^{4}}+2x+2y+3y+2{{y}^{4}}$
3. $l\displaystyle ({{y}^{4}}+2{{y}^{4}})+2x+(2y+3y)=$
$\displaystyle 3{{y}^{4}}+2x+5y=$
4. We don´t have constants to combine, so we simplified our expression with 3 steps.
Algebraic Formulas
The basic algebraic formulas we use when we want to simplify or solve an equation are:
Square Formulas
Cube Formulas
Example 5: Simplify the given expression by following the proper steps and write the type of the expression.
a) $\displaystyle {{(x+3)}^{2}}+2{{x}^{2}}+12-21$
b) $\displaystyle 3x-2+{{(2y-3)}^{2}}+{{({{x}^{2}})}^{{-2}}}$
Solution
a) Remove the parenthesis by expanding the “Square Formula”
$\displaystyle {{(x+3)}^{2}}+2{{x}^{2}}+12-21=$
$\displaystyle {{x}^{2}}+2\cdot 3\cdot x+{{(3)}^{2}}+2{{x}^{2}}+12-21=$
$ \displaystyle {{x}^{2}}+6x+9+2{{x}^{2}}+12-21=$
Combine the like terms of the expression
$ \displaystyle ({{x}^{2}}+2{{x}^{2}})+6x+(9+12-21)=$
$ \displaystyle 3{{x}^{2}}+6x+0=$
$\displaystyle 3{{x}^{2}}+6x$
The type of the expression is Binomial Expression.
b) Remove the parenthesis by expanding “Square Formula”
$\displaystyle 3x-2+{{(2y-3)}^{2}}+{{({{x}^{2}})}^{{-2}}}$
$\displaystyle 3x-2+{{(2y)}^{2}}-2\cdot 2y\cdot 3+{{(-3)}^{2}}+{{({{x}^{2}})}^{{-2}}}=$
$\displaystyle 3x-2+4{{y}^{2}}-12y+9+{{({{x}^{2}})}^{{-2}}}=$
Secondly we use the exponent rule by multiplying the exponents.
$\displaystyle 3x-2+4{{y}^{2}}-12y+9+{{x}^{{-4}}}=$
$\displaystyle 3x-2+4{{y}^{2}}-12y+9+\frac{1}{{{{x}^{4}}}}=$
Combine the like terms of the expression
$\displaystyle 3x+4{{y}^{2}}-12y+\frac{1}{{{{x}^{4}}}}+(9-2)=$
$ \displaystyle 3x+4{{y}^{2}}-12y+\frac{1}{{{{x}^{4}}}}+7=$
The type of the expression is a Polynomial Expression.
