How to Write Expressions and Equations in Standard Algebraic Form
The standard algebraic format deals with the symbolic presentation of any valid mathematical statement in terms of some specific algebraic expressions, identities, or equations. Here, the different combinations of mathematical operations like division, multiplication, addition, subtraction, and exponentiation are executed about the conditions of the problems. The mathematical operators follow a specific " BODMAS " sequence to arrive at the correct answer to any algebraic expression.
What are the Parameters Associated with an Algebraic Format?
The following parameters associated with an algebraic format may be present in algebraic expressions, identities, or equations.
Constants: A quantity that has a defined value and does not change throughout the operations done in the algebraic format. Examples are the numerical values, or any letters assumed to be constants.
Variables: A variable is the symbolic representation of an unknown value. We usually represent the variables using letters such as x, y, or t. For instance, when we declare that l stands for the length of a rectangle and w stands for the width of the rectangle; we can represent its area A as $A=l\times w$.
Coefficients: The coefficient is the constant that accompanies the variable.
For example, in the entity $3 \times y$ or$3y$, 3 is the coefficient of the variable y.Terms: All the individual entities of an algebraic expression, identity, or equation, which are separated by the operators, are commonly known as the terms. Thus, a term may be a constant or a product of a coefficient and a variable.
In the algebraic equation $x+3=3{{y}^{2}}+2z$, there are four terms, namely, $x$, $3$, $3{{y}^{2}}$, $2z$.
What are Algebraic Expressions?
The mathematical expression containing one or more “algebraic variables” is called an algebraic expression. It may be along with the associated “coefficients” and “constants”. Examples of algebraic expressions are $a{{x}^{2}}+b$, ${{y}^{3}}+3$,
When the “equal to sign” symbolised as “=” is used to unite any two or more algebraic expressions, the resulting mathematical entity is called an “algebraic equation”. An algebraic equation may correlate an algebraic expression to any constant only. It suggests that the terms or expressions on either side of the “=” are equivalent.
Examples of algebraic equations are $a{{x}^{2}}+b=c$, $ {{y}^{3}}=3$.
Terms of Equation
What are the Types of the Algebraic Equations?
According to the number of terms used to form an algebraic expression, some of the algebraic equations are explained below.
Monomial: An algebraic expression having just one term is called a monomial. for instance, etc. $3y,2xyz,4x,-xy,\dfrac{{-5}}{3}abc,$, are monomials.
Binomial: An algebraic expression that contains two unlike terms is named a binomial. for instance , $x + y,4p + 2z,3{x^2}-{y^2}$etc., are binomials.
Trinomial: An algebraic expression having three unlike terms is named a trinomial. for instance , $a + 4b + 2z,x-pq + yz,$etc., are trinomials.
Quadrinomial: An algebraic expression containing four terms is named a quadrinomial. for instance , $7ab-c + z + 4xy,abc-a-b-c$ are quadrinomials.
Multinomial: An algebraic expression with two or quite two terms is called a multinomial. for instance, $4x + 3,5-x,{y^2} + 7y$, each may be a multinomial of two terms is a multinomial of four terms. $-ab + 7b-6bc-z,a-ab + 7b-6bc-z$a may be a multinomial of five terms, and so on.
Polynomial: An algebraic expression with one or more unlike terms with the facility of the variables as only whole numbers is called a polynomial. In other words, all monomials, binomials, trinomials, and every other expression having any number of finite terms with the power of their variables as whole numbers are called polynomials. No term during a polynomial contains a negative exponent or any variable in the denominator. for instance $4{{x}^{3}}+3{{x}^{2}}+9x-1,9{{a}^{2}}+5,4{{x}^{2}}+3{{x}^{2}}+9x-1,9{{a}^{2}}+5$, are polynomials.
What are Algebraic Identities?
The identities dealing with only the algebraic expressions that may contain coefficients of algebraic variables and constants are known as “algebraic identities”.
Algebraic identities are always true for any value of the algebraic variables involved in one or both the expressions on either side of the “=”.
The mathematical identity ${{(a+b)}^{2}}={{a}^{2}}+{{b}^{2}}+2ab$ is an example of an algebraic identity.
What are the Advantages of Algebraic Identities?
The following are some of the advantages of algebraic identities.
An algebraic identity does not show a finite number of solutions, as it is true for an infinite number of values you assign to its variables.
The algebraic identities are useful in the simplification of algebraic polynomials.
The algebraic identities may be utilised in the factorisation of algebraic expression
The algebraic identities also help solve any algebraic equation.
What are Algebraic Polynomials?
Algebraic polynomials are expressions or equations possessing only the non-negative integral power of variables of the algebraic expressions or equations but may contain any integral coefficients and constants.
Following are examples of algebraic polynomials.
${{(a+b)}^{3}}, {{a}^{2}}+{{b}^{2}}+2ab, {{x}^{2}}+{{y}^{2}}+2\sqrt{3}$
What are the Types of Algebraic Polynomials in One Variable?
By the degree of the polynomial, the following are common types of polynomials in one variable.
1. Linear Polynomial
This polynomial is of degree $1$. The standard form of the linear polynomial is ${{a}_{0}}x+{{a}_{1}}\text{;}$
$\text{where }{{a}_{n}}\text{ is a real number;} $
$ n=0,1\text{ ;} $
$ {{a}_{0}}\ne 0$
2. Quadratic Polynomial
This polynomial is of degree $2$. The standard form of the quadratic polynomial is
$ {{a}_{0}}{{x}^{2}}+{{a}_{1}}x+{{a}_{2}}\text{; }$
$ \text{where }{{a}_{n}}\text{ is a real number;}$
$n=0,1,2\text{ ;}$
${{a}_{0}}\ne 0$
3. Cubic Polynomial
This polynomial is of degree $3$. The standard form of the cubic polynomial is ${{a}_{0}}{{x}^{3}}+{{a}_{1}}{{x}^{2}}+{{a}_{2}}x+{{a}_{3}};$
$\text{where }{{a}_{n}}\text{ is a real number;}$
$ n=0,1,2,3\text{ ;}$
$ {{a}_{0}}\ne 0 $
Chart of Useful Algebraic Identities
The following table shows some of the important algebraic identities.
Interesting Facts
The mathematical identity ${{\cos }^{2}}\theta +{{\sin }^{2}}\theta =1$ is not an “algebraic identity” but a “trigonometric identity”.
The algebraic identities may be conditional also. For example, ${{a}^{3}}+{{b}^{3}}+{{c}^{3}}=3abc,\quad$ when $ a+b+c=0$
The mathematical expression $\sqrt{3}+2$ is not an “algebraic expression” but a numeric expression.
Solved Examples
1. Find the value of $\left( {x{\rm{ }} + {\rm{ }}1} \right)\left( {x{\rm{ }} + {\rm{ }}1} \right)$ using standard algebraic identities.
Ans. Note that $\left( {x{\rm{ }} + {\rm{ }}1} \right)\left( {x{\rm{ }} + {\rm{ }}1} \right)$ is equivalent to ${\left( {x{\rm{ }} + {\rm{ }}1} \right)^2}$.
Thus, by using the Identity applicable for the whole square of the sum of any two variables ${{(x+y)}^{2}}={{x}^{2}}+{{y}^{2}}+2xy$ and substituting here $y=1$, we get the following.
${{\left( x+1 \right)}^{2}}\ ={{x}^{2}}\ +2\times x\times 1+{{1}^{2\;}}={{x}^{2\;}}+2x+1$
2. Factorize $\left( {{x^4}\;-{\rm{ }}1} \right)$ using standard algebraic identities.
Ans. Split $\left( {{x^4}\;-{\rm{ }}1} \right)$using the Identity for the difference between any two squares
Replace here x by and put y = 1.
$\left( {{x^4}\;-{\rm{ }}1} \right){\rm{ }} = {\rm{ }}\left( {{{\left( {{x^2}} \right)}^2}-{\rm{ }}{1^2}} \right){\rm{ }} = {\rm{ }}\left( {{x^{2\;}} + {\rm{ }}1} \right)\left( {{x^{2\;}}-{\rm{ }}1} \right)$
The factor $(x^2 -1)$ is further factorized using the same Identity where y = 1, and we obtain the following
$\left( {{x^4}\;-{\rm{ }}1} \right){\rm{ }} = {\rm{ }}\left( {{x^{2\;}} + {\rm{ }}1} \right)\left( {{{\left( x \right)}^{2\;}}-{{\left( 1 \right)}^2}} \right){\rm{ }} = {\rm{ }}\left( {{x^{2\;}} + {\rm{ }}1} \right)\left( {x{\rm{ }} + {\rm{ }}1} \right)\left( {x{\rm{ }}-{\rm{ }}1} \right)$
3. Expand ${\left( {3x{\rm{ }}-{\rm{ }}4y} \right)^{3\;}}$using standard algebraic identities.
Ans: We can expand ${\left( {3x{\rm{ }}-{\rm{ }}4y} \right)^{3\;}}$using the identity applicable to the whole cube of the difference of any two variables ${{(x-y)}^{3}}={{x}^{3}}-3{{x}^{2}}y+3x{{y}^{2}}-{{y}^{3}}$
Replacing x and y by $3x$ and $4y$, we get the following.
${{\left( 3x-4y \right)}^{3}}\ ={{\left( 3x \right)}^{3}}-3{{\left( 3x \right)}^{2}}\left( 4y \right)+3\left( 3x \right){{\left( 4y \right)}^{2}}-{{\left( 4y \right)}^{3}}=27{{x}^{3}}\ -108{{x}^{2}}y+144x{{y}^{2}}\ -64{{y}^{3}}$
4. Factorize ${\rm{ }}({x^3}\; + {\rm{ }}8{y^{3\;}} + {\rm{ }}27{z^3}\;-{\rm{ }}18xyz)$using standard algebraic identities.
Ans. The expression ${\rm{ }}({x^3}\; + {\rm{ }}8{y^{3\;}} + {\rm{ }}27{z^3}\;-{\rm{ }}18xyz)$ can be factorized using the identity \[\left( {{x}^{3}}\ +{{y}^{3\;}}+{{z}^{3}}\ -3xyz \right)=\left( x+y+z \right)\left( {{x}^{2\;}}+{{y}^{2}}\ +{{z}^{2}}\ -xy-yz-zx \right)\]
Therefore, we get the following
$\left( {{x^3}\; + {\rm{ }}8{y^{3\;}} + {\rm{ }}27{z^3}\;-{\rm{ }}18xyz} \right){\rm{ }} = {\rm{ }}{\left( x \right)^3}\; + {\rm{ }}{\left( {2y} \right)^{3\;}} + {\rm{ }}{\left( {3z} \right)^3}\;-{\rm{ }}3\left( x \right)\left( {2y} \right)\left( {3z} \right) = {\rm{ }}\left( {x{\rm{ }} + {\rm{ }}2y{\rm{ }} + {\rm{ }}3z} \right)\left( {{x^{2\;}} + {\rm{ }}4{y^2}\; + {\rm{ }}9{z^2}\;-{\rm{ }}2xy{\rm{ }}-{\rm{ }}6yz{\rm{ }}-{\rm{ }}3zx} \right)$
Summary
The standard algebraic format encompasses the algebraic expressions, the algebraic identities, and the algebraic equations.
The algebraic expressions, the algebraic identities, and the algebraic equations can be of various types varying from monomials to polynomials.
The algebraic expressions, the algebraic identities, and the algebraic equations can be linear, quadratic, cubic, or of any higher degree.
The algebraic identities find usages as algebraic formulae in various mathematical problems.
Practice Problems
1. Find the value of ${{\left( a+2 \right)}^{3}}$.
2. Find the value of ${{\left( a+2+3x \right)}^{2}}$.
3. Find the value of $729{{x}^{3}}-216{{y}^{3}}$
FAQs on Understanding Standard Algebraic Formats in Algebra
1. What is standard algebraic form?
The standard algebraic form is a way of writing algebraic expressions or equations in a consistent, simplified, and ordered manner. In most cases, this means:
- Terms are arranged in descending powers (highest exponent first).
- Like terms are combined.
- The expression is written clearly without unnecessary brackets.
For example, 3 + 2x − x² written in standard form is −x² + 2x + 3.
2. What is the standard form of a linear equation?
The standard form of a linear equation in two variables is Ax + By = C, where A, B, and C are constants and A and B are not both zero. This form is commonly used in coordinate geometry.
- A, B, and C are usually integers.
- The variables x and y are written on the left side.
- The constant is written on the right side.
Example: 2x + 3y = 6 is in standard form.
3. What is the standard form of a quadratic equation?
The standard form of a quadratic equation is ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0. This format is required to apply the quadratic formula.
- a is the coefficient of x²
- b is the coefficient of x
- c is the constant term
Example: 2x² − 5x + 3 = 0 is in standard quadratic form.
4. How do you write an expression in standard algebraic form?
To write an expression in standard algebraic form, arrange terms in descending powers and combine like terms. Follow these steps:
- Expand brackets if necessary.
- Combine like terms (same variables and exponents).
- Arrange terms from highest exponent to lowest.
Example: 4x − 2 + 3x² + x becomes 3x² + 5x − 2.
5. What is the difference between standard form and expanded form?
The difference is that standard form arranges terms in descending powers, while expanded form removes brackets by multiplying them out. Expanded form focuses on removing parentheses, whereas standard form focuses on order and simplification.
- Expanded form example: (x + 2)(x + 3) = x² + 5x + 6
- Standard form of the result: x² + 5x + 6
Often, an expression is expanded first and then written in standard form.
6. Why is standard algebraic form important?
Standard algebraic form is important because it makes equations easier to solve, compare, and apply formulas correctly. Many algebraic methods require expressions in standard form.
- The quadratic formula needs ax² + bx + c = 0.
- Graphing linear equations often uses Ax + By = C.
- It helps identify coefficients and constants quickly.
Using standard form reduces mistakes and improves clarity in algebra.
7. Can you give an example of converting a quadratic to standard form?
Yes, converting a quadratic to standard form means rearranging it into ax² + bx + c = 0. Example:
- Given: x(2x − 3) = 5
- Expand: 2x² − 3x = 5
- Move all terms to one side: 2x² − 3x − 5 = 0
The standard form is 2x² − 3x − 5 = 0.
8. What is the standard form of a polynomial?
The standard form of a polynomial arranges all terms in descending order of exponents and combines like terms. This format clearly shows the degree of the polynomial.
- Highest exponent first
- Constant term last
- No repeated like terms
Example: 7 + 4x³ − x + 2x² becomes 4x³ + 2x² − x + 7.
9. What is the standard form of a number in algebra?
In algebra, the standard form of a number often refers to scientific notation written as a × 10ⁿ, where 1 ≤ a < 10 and n is an integer. This is used for very large or very small numbers.
- Example: 4500 = 4.5 × 10³
- Example: 0.006 = 6 × 10⁻³
This format makes calculations and comparisons easier.
10. What are common mistakes when writing standard algebraic form?
Common mistakes in writing standard algebraic form include not arranging terms correctly and failing to combine like terms. Avoid these errors:
- Leaving terms unordered (e.g., 3 + x² − x).
- Not combining like terms (e.g., 2x + 3x).
- Forgetting to move all terms to one side in equations.
For example, 2x + 3x + 1 should be written as 5x + 1 in standard form.
