Question

classify each polynomial as constant, linear, quadratic, or cubic. combine like terms first.

\\(x^2 - 2x + x^3\\)
\\(4x^2 - 6x - 8x^2\\)
\\(6x - 6 + 6x\\)
\\(5 + 4x^2 - 4x^2 + 5\\)

classify each polynomial as a monomial, binomial, or trinomial. combine like terms first.

\\(x^3 + 3x^3 + 2x\\)
\\(2x^3 + 5x + 3x^4 - x\\)
\\(4x - 5x + x - 2\\)
\\(6x^2 + 5 - 2x^2 - 9\\)

Explanation:

🆕 New Concept Discovered: Classifying Polynomials
Categorizing expressions by degree and number of terms.

Step 1: Classify by Degree (Left Column)

To classify a polynomial by degree (constant, linear, quadratic, or cubic), we first combine any like terms and find the highest exponent (degree) of the variable \( x \):

• Constant: Degree 0 (no variable, e.g., \( 10 \))
• Linear: Degree 1 (e.g., \( x \))
• Quadratic: Degree 2 (e.g., \( x^2 \))
• Cubic: Degree 3 (e.g., \( x^3 \))

Let's simplify and classify each expression:

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

• No like terms to combine.
• Highest exponent is 3.
• Classification: cubic

1. \( 4x^2 - 6x - 8x^2 \)

• Combine like terms: \( (4x^2 - 8x^2) - 6x = -4x^2 - 6x \)
• Highest exponent is 2.
• Classification: quadratic

1. \( 6x - 6 + 6x \)

• Combine like terms: \( (6x + 6x) - 6 = 12x - 6 \)
• Highest exponent is 1.
• Classification: linear

1. \( 5 + 4x^2 - 4x^2 + 5 \)

• Combine like terms: \( (4x^2 - 4x^2) + (5 + 5) = 0x^2 + 10 = 10 \)
• No variable remains (degree 0).
• Classification: constant

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Step 2: Classify by Number of Terms (Right Column)

To classify a polynomial by the number of terms (monomial, binomial, or trinomial), we must combine all like terms first and then count the remaining non-zero terms:

• Monomial: 1 term
• Binomial: 2 terms
• Trinomial: 3 terms

Let's simplify and classify each expression:

1. \( x^3 + 3x^3 + 2x \)

• Combine like terms: \( (1x^3 + 3x^3) + 2x = 4x^3 + 2x \)
• There are 2 terms remaining.
• Classification: binomial

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

• Combine like terms: \( 3x^4 + 2x^3 + (5x - x) = 3x^4 + 2x^3 + 4x \)
• There are 3 terms remaining.
• Classification: trinomial

1. \( 4x - 5x + x - 2 \)

• Combine like terms: \( (4x - 5x + x) - 2 = 0x - 2 = -2 \)
• There is 1 term remaining.
• Classification: monomial

1. \( 6x^2 + 5 - 2x^2 - 9 \)

• Combine like terms: \( (6x^2 - 2x^2) + (5 - 9) = 4x^2 - 4 \)
• There are 2 terms remaining.
• Classification: binomial

Answer:

Left Column (Classify as constant, linear, quadratic, or cubic):

• \( x^2 - 2x + x^3 \) \(

ightarrow\) cubic

• \( 4x^2 - 6x - 8x^2 \) \(

ightarrow\) quadratic

• \( 6x - 6 + 6x \) \(

ightarrow\) linear

• \( 5 + 4x^2 - 4x^2 + 5 \) \(

ightarrow\) constant

Right Column (Classify as monomial, binomial, or trinomial):

• \( x^3 + 3x^3 + 2x \) \(

ightarrow\) binomial

• \( 2x^3 + 5x + 3x^4 - x \) \(

ightarrow\) trinomial

• \( 4x - 5x + x - 2 \) \(

ightarrow\) monomial

• \( 6x^2 + 5 - 2x^2 - 9 \) \(

ightarrow\) binomial