# Standard Form

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## What is Standard Form?

A standard form is a method of writing mathematical concepts like an equation, expressions, or numbers in standard form.

Example:

2.5 billion years written as 2,500,000,000 years.

As you can see, reading or writing a large number like 2.5 billion is not only difficult but also time-consuming and there are chances of writing more or less 0’s while writing the larger numbers in a general form.

Hence, to write large or small numbers precisely, we use the standard form.

### Standard Form Formula

The method of writing mathematical concepts like an equation, expressions, or numbers in standard form follows certain rules or formulas.

The standard form formulas vary for different Mathematics concepts.

For example, the standard form of 2,500,000,000 is 2.5 x 10⁹

What about the fractions 12/24 and 15/7? Are these standard forms of fractions?

In the case of fractions, we need to ensure that in the standard form of decimals, both numerator and denominators must be coprime.

It implies that they should not have common factors other than 1.

So, the standard form of fraction 12/24 is 1/2

The fraction 15/7 is already written in a standard form, as both 15 and 7 are co-prime.

### Polynomial

A polynomial is defined as a mathematical expression that includes variables, coefficients, and operations of addition, subtraction, multiplication, and non - negative integer.

The standard of a polynomial of degree nis anxn………+ a1x+a0

Examples: x + 2, 3y2 - 2y + 5, -6, 1/2y2 - 2/3 y + 3/4

### What Does Standard Form of a Polynomial Means

The standard form of a polynomial is a method of writing polynomials with the exponents in decreasing order.

Polynomials are expressed in standard form to make the complex calculation easier.

A polynomial is considered to be written in standard form, if it is expressed in such a way that the term with the highest degree is written first, followed by the term which has the next highest degree, and so on.

Example : 14y4 - 5y3 - 11y2 - 11y + 8

You can find that in the above - given standard form of a polynomial, the exponents are placed in decreasing order.

### How to Write Polynomials In Standard Form?

The like terms in the standard form of a polynomial are grouped, added, subtracted, and rearranged with the exponents of terms in decreasing order.

Following are the steps to write a polynomial in standard form:

1. Write the terms.

2. Arrange all the like terms.

3. Find the exponent.

4. Write the term with the highest exponent first.

5. Write the remaining terms with lower exponents in decreasing order

6. Write the constant term ( a number without variable) in the end.

Example: 8y⁴ + 11y³ - 6y⁴ - 8y²

= 8y⁴ - 6y⁴ + 11y³ + 8y²

= 2y³ + 11y³ + 8y²

In the above example, the polynomial with the highest degree is 4 and that became the exponents with the first term.

### Standard Form of an Equation

The standard form of an equation is written as  Ax + By = C, where A, B, and C are integers.

This form of the equation is particularly useful for determining both the x - and y-intercepts. We can find the x-intercept of an equation by substituting 0 for y and solving for the x. Similarly, we can find the y-intercept of an equation by substituting 0 for x and solving for y.

Example: Put y² = 6 in standard form.

Answer: The equation y² = 6 in standard form can be written as”

y² - 6 = 0

### Standard Form of a Decimal Number

The standard form of a decimal number in Britain is known as Scientific notation, where the  number is written in the following way:

4527.7                =  4.5277 10³

A Number              In Scientific Notation

In this example, 4527.7 can be written as 4.5277 × 10³ in scientific notation

Because, 4527.7 =  4.5277 × 1000 = 4.5277 × 10³

The standard form of decimal numbers in the United States, and in other countries using the US conventions is written in expanded form.

Example: 4.327 in expanded form can be written as:

$4.327 = 4 \times 1 + 3 \times \frac{1}{10} + 2 \times \frac{1}{100} + 7 \times \frac{1}{1000}$

4.327 = 4 + 0.3 + 0.03 + 0.007

Therefore,

The expanded form of 4.327 is 4 + 0.3 + 0.03 + 0.007.

### Standard Form Examples

Here are some standard form examples for different mathematical concepts.

1. Write the polynomial y² - 10y + 16 - y² + y⁵ - 3y⁴ + 3y²

Solution:

To write the given polynomial equation in standard form, two rules should be followed.

1. Write the terms in decreasing order of their powers.

2. All the terms should be unlike.

Let us first arrange the given term in decreasing order:

y² - 10y + 16 - y² + y⁵ - 3y⁴ + 3y² = y⁵ - 3y⁴ + y² - y² + 3y² -10y + 16

After adding like terms, we get

y⁵ - 3y⁴  + 3y² -10y + 16

Hence, the standard form is y⁵ - 3y⁴  + 3y² - 10y + 16

2. How to write the distance between Sun and Mars in standard form?

Solution:

As we know, the distance between the Sun and Mars is 228,000,000 km or 141,700,000 miles.

Hence, the distance between the Sun and Mars in standard form can be written as or 2.28 × 108 km or 1.417 × 108 miles.

3. Write 3253 in standard form.

Solution:

3253 can be written as 3.253 × 1000

Therefore, the standard form of 3253 is 3.253 × 10³

Q1. Who Introduced Standard Form in Mathematics?

Ans. Standard form in Mathematics was introduced by the Persian Mathematician in the 9th Century named Muhammad-Al Khwarizimi.

Q2. What is the Meaning of Scientific Notation?

Ans. Scientific notation is the method of writing numbers so that they can be easily evaluated and manipulated. The method is standardised to be easy to read, with a one single-digit number written before the decimal point, and the exponents showing the overall intensity of a number.

Scientific notation provides techniques to avoid mistakes that come with calculations that include very large or small numbers.

Q3. How Scientific Notation is Helpful?

Ans.

• The exponents written in scientific notation can be helpful to compare two large numbers quickly and accurately. For example, you can easily compare which number is larger between 3 × 108 and 3 × 1011, than to make a comparison between 200000000 and 200000000000.

• Most of the zeros written in a number such as 5023 000 000 000 000 000 000 are completely meaningless. This can be written precisely in  standard form. Writing the number as 5.023 × 1021 shows that precision is only to four significant figures.