Definition of Standard Form

Standard form is the standard way of writing something. In mathematics, there are many different standard forms. For example, there is a standard form for writing a number, an equation, a polynomial, and so forth.

We will go through the most common algebraic standard forms in this lesson.

Standard Form of a Number

Writing numbers in standard form is also known as scientific notation. The form is given as:

C x 10ⁿ

Where C is a decimal number between 0 and 10, such as 6.35. C is shown with two decimal places in the following example. However, we can write out as many decimal places as necessary.

Example problem:

Write the number 36,500 in standard form, also known as scientific notation.

Solution:

1. Let's move the decimal to the left 4 places so that C = 3.65. C will need to be multiplied by 10⁴ to equal 36,500.
2. The number is written in standard form as 3.65 x 10⁴.

Standard Form of a Polynomial

A polynomial is in standard form if the terms' exponents are in descending order. In other words, the term with the highest power should be furthest to the left and the term with the lowest power should be furthest to the right.

Example problem:

Rewrite the polynomial 3x² + x - 2x³ + 6 in standard form.

Solution:

1. Let's rearrange the terms so that the exponents are in descending order.
2. 3x² + x - 2x³ + 6 → -2x³ + 3x² + x + 6
3. The polynomial is rewritten in standard form as -2x³ + 3x² + x + 6.

Standard Form of an Equation

An equation is in standard form if it is set equal to zero. For example, the equation 4y³ + y = 5 can be rearranged to standard form by moving the 5 to the left side of the equals sign. Standard form of this equation is 4y³ + y - 5 = 0.

Standard Form of a Linear Equation

The standard form of a linear equation is given as:

Ax + By = C

Where A, B, and C are constants, and x and y are variables. This standard form is only applicable for the equation of a line.

Example problem:

Rewrite the equation y = 4x + 2 in standard form.

Solution:

1. The equation is currently in slope intercept form. Let's move the term 4x over to the left side of the equals sign.
2. y = 4x + 2 → -4x + y = 2
3. The linear equation is written as -4x + y = 2 in standard form.

Standard Form of a Quadratic Equation

The standard form of a quadratic equation is given as:

ax² + bx + c = 0

Where a, b, and c, are constants, and x is a variable.

Example problem:

Rewrite the quadratic -4x - 12 = 2x² in standard form.

Solution:

1. We must have all nonzero terms on one side of the equals, and the zero term on the other side of the equals. The nonzero terms must also be in descending order of exponent power.
2. -4x - 12 = 2x² → 2x² + 4x + 12 = 0
3. The quadratic is written as 2x² + 4x + 12 = 0 in standard form.