Question

which of the following polynomials is in standard form?

( ) (2x + 5x^3 - 1)
( ) (5x^3 + 2x - 1)
( ) (-1 + 2x + 5x^3)
( ) (2x - 1 + 5x^3)

Explanation:

Define standard form of a polynomial

A polynomial is written in standard form when its terms are ordered from the highest degree (exponent) to the lowest degree.

Analyze the degrees of each term

Using the Polynomial Classification knowledge point
The terms in the given expressions are \(5x^3\), \(2x\), and \(-1\).

• The degree of \(5x^3\) is 3.
• The degree of \(2x\) is 1.
• The degree of \(-1\) is 0.

Order the terms from highest to lowest degree

To write the polynomial in standard form, we arrange the terms in descending order of their exponents:

$$ 3 > 1 > 0 $$

This gives the ordered sequence:

$$ 5x^3 + 2x - 1 $$

Match with the given options

We compare our ordered polynomial with the choices:

• \(2x + 5x^3 - 1\) (Degrees: 1, 3, 0)
• \(5x^3 + 2x - 1\) (Degrees: 3, 1, 0)
• \(-1 + 2x + 5x^3\) (Degrees: 0, 1, 3)
• \(2x - 1 + 5x^3\) (Degrees: 1, 0, 3)

The second option is in standard form.

Answer:

• (A) \(2x + 5x^3 - 1\)
• (B) \(5x^3 + 2x - 1\) (Correct answer)
• (C) \(-1 + 2x + 5x^3\)
• (D) \(2x - 1 + 5x^3\)