Question

polynomial classification - practicetwrite in standard form(1) 9 + 5x² - 3x - 2x³(2) 9x⁹ + 5x² - 8 - 3x - 2x¹¹(3) 9w + 5w² - 3w⁴ - 2w³(4) 9 - x

Explanation:

Problem (1): \( 9 + 5x^2 - 3x - 2x^3 \)

Step 1: Recall Standard Form

A polynomial in standard form is written with terms in descending order of their exponents.

Step 2: Identify Exponents

For the polynomial \( 9 + 5x^2 - 3x - 2x^3 \), the exponents of \( x \) in each term are:

• For \( -2x^3 \): exponent is \( 3 \)
• For \( 5x^2 \): exponent is \( 2 \)
• For \( -3x \): exponent is \( 1 \) (since \( x = x^1 \))
• For \( 9 \): exponent is \( 0 \) (since \( 9 = 9x^0 \))

Step 3: Rearrange Terms

Rearrange the terms in descending order of their exponents:
\( -2x^3 + 5x^2 - 3x + 9 \)

Step 1: Recall Standard Form

A polynomial in standard form has terms ordered by descending exponents.

Step 2: Identify Exponents

For the polynomial \( 9x^9 + 5x^2 - 8 - 3x - 2x^{11} \), the exponents of \( x \) are:

• For \( -2x^{11} \): exponent is \( 11 \)
• For \( 9x^9 \): exponent is \( 9 \)
• For \( 5x^2 \): exponent is \( 2 \)
• For \( -3x \): exponent is \( 1 \)
• For \( -8 \): exponent is \( 0 \) (since \( -8 = -8x^0 \))

Step 3: Rearrange Terms

Rearrange the terms in descending order of exponents:
\( -2x^{11} + 9x^9 + 5x^2 - 3x - 8 \)

Step 1: Recall Standard Form

Standard form of a polynomial in \( w \) is terms in descending order of \( w \)'s exponents.

Step 2: Identify Exponents

For the polynomial \( 9w + 5w^2 - 3w^4 - 2w^3 \), the exponents of \( w \) are:

• For \( -3w^4 \): exponent is \( 4 \)
• For \( -2w^3 \): exponent is \( 3 \)
• For \( 5w^2 \): exponent is \( 2 \)
• For \( 9w \): exponent is \( 1 \) (since \( w = w^1 \))

Step 3: Rearrange Terms

Rearrange the terms in descending order of exponents:
\( -3w^4 - 2w^3 + 5w^2 + 9w \)

Answer:

\( -2x^3 + 5x^2 - 3x + 9 \)

Problem (2): \( 9x^9 + 5x^2 - 8 - 3x - 2x^{11} \)