Question

which polynomial is in standard form?

\\(1 - 2x - 8x^2 + 8x^3\\)

\\(2x^2 + 8x^3 - 8x + 12\\)

\\(8x^3 + 5x - 3x^2 + 2\\)

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

Explanation:

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

Step 1: Define Standard Form

A polynomial is in standard form when its terms are written in descending order of their exponents (degrees), from the highest exponent to the lowest exponent.

For a single variable \( x \), the terms should follow the pattern:

$$ x^n, x^{n-1}, \dots, x^1, \text{constant} $$

Step 2: Analyze the Options

Let's look at the exponents of each term from left to right for each given option:

• Option 1: \( 1 + 2x - 8x^2 + 8x^3 \)
• Exponents: \( 0, 1, 2, 3 \) (This is in ascending order, not descending).
• Option 2: \( 2x^3 + 8x^2 - 8x + 12 \)
• Exponents: \( 3, 2, 1, 0 \) (This is in descending order).
• Option 3: \( 8x^3 + 5x - 3x^2 + 2 \)
• Exponents: \( 3, 1, 2, 0 \) (Not in order because \( 1 \) is followed by \( 2 \)).
• Option 4: \( 2x^3 + 4x^2 - 7x + 5 \)
• Exponents: \( 3, 2, 1, 0 \) (This is also in descending order).

Note on image transcription: Looking closely at the image, the last option is \( 2x^3 + 4x^2 - 7x + 5 \), which perfectly follows the descending order of powers: \( 3
ightarrow 2
ightarrow 1
ightarrow 0 \).

Answer:

The polynomial in standard form is:

$$ 2x^3 + 4x^2 - 7x + 5 $$