Question

which polynomial is in standard form?

• (3 + x)
• (x^7 + 2x^6 - 3x^2 + 6x^5 - 3x^4 + 4x + 8)
• (8a^3 + 10a - 12a^2)
• (-2x^6 + x^5 - 3x^4 + 7x^3 - 2x^2 + 3x + 1)

Explanation:

🆕 New Concept Discovered: Polynomial Standard Form
Arranging terms from highest to lowest exponent.

Step 1: Define Standard Form

A polynomial is in standard form when its terms are written in descending order of their degrees (exponents). This means we start with the term that has the largest exponent and move to the term with the smallest exponent, ending with any constant term.

Step 2: Analyze the Options

We will check the exponents of each term from left to right for each given option:

• Option 1: \( 3 + x \)
• The exponents are \( 0 \) (for the constant \( 3 \)) and \( 1 \) (for \( x^1 \)).
• This is in ascending order, not descending.

• Option 2: \( x^7 + 2x^6 - 3x^2 + 6x^5 - 3x^4 + 4x + 8 \)
• The exponents are \( 7, 6, 2, 5, 4, 1, 0 \).
• This is not in descending order because \( 2 \) comes before \( 5 \).

• Option 3: \( 8a^3 + 10a - 12a^2 \)
• The exponents are \( 3, 1, 2 \).
• This is not in descending order because \( 1 \) comes before \( 2 \).

• Option 4: \( -2x^6 + x^5 - 3x^4 + 7x^3 - 2x^2 + 3x + 1 \)
• The exponents are \( 6, 5, 4, 3, 2, 1, 0 \).
• This is in perfect descending order.

Answer:

\( -2x^6 + x^5 - 3x^4 + 7x^3 - 2x^2 + 3x + 1 \)