Question

classify the following polynomials according to the number of terms. combine any like terms first.

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

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

\\(-4x^2 + x + x^2 - 2x\\)

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

Explanation:

🆕 New Concept Discovered: Classifying Polynomials
Naming polynomials by their number of terms

Step 1: Understand the classification rules

A polynomial is classified by the number of non-zero, non-like terms it has after simplifying:

• Monomial: 1 term (e.g., \(3x\))
• Binomial: 2 terms (e.g., \(x^2 + 5\))
• Trinomial: 3 terms (e.g., \(x^2 + 3x - 2\))

We must combine any like terms (terms with the same variables raised to the same powers) before counting.

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Step 2: Analyze the first polynomial

The first expression is:
\[ x^2 - 3x + 2x \]

Combine the like terms \(-3x\) and \(2x\):
\[ x^2 + (-3x + 2x) = x^2 - x \]

This simplified expression has 2 terms (\(x^2\) and \(-x\)). Therefore, it is a binomial.

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Step 3: Analyze the second polynomial

The second expression is:
\[ x^2 + x + 2x^2 - x \]

Group and combine the like terms:

• For \(x^2\) terms: \(x^2 + 2x^2 = 3x^2\)
• For \(x\) terms: \(x - x = 0\)

The simplified expression is:
\[ 3x^2 \]

This simplified expression has 1 term. Therefore, it is a monomial.

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Step 4: Analyze the third polynomial

The third expression is:
\[ 4x^2 + x + x^2 - 2x \]

Group and combine the like terms:

• For \(x^2\) terms: \(4x^2 + x^2 = 5x^2\)
• For \(x\) terms: \(x - 2x = -x\)

The simplified expression is:
\[ 5x^2 - x \]

This simplified expression has 2 terms. Therefore, it is a binomial.

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Step 5: Analyze the fourth polynomial

The fourth expression is:
\[ 3x^2 + x - 3x^2 - x \]

Group and combine the like terms:

• For \(x^2\) terms: \(3x^2 - 3x^2 = 0\)
• For \(x\) terms: \(x - x = 0\)

The simplified expression is:
\[ 0 \]

Since all terms cancel out, this simplifies to a single constant value of \(0\), which is a single term. Therefore, it is a monomial.

Answer:

• \(x^2 - 3x + 2x\) \(

ightarrow\) binomial (simplifies to \(x^2 - x\))

• \(x^2 + x + 2x^2 - x\) \(

ightarrow\) monomial (simplifies to \(3x^2\))

• \(4x^2 + x + x^2 - 2x\) \(

ightarrow\) binomial (simplifies to \(5x^2 - x\))

• \(3x^2 + x - 3x^2 - x\) \(

ightarrow\) monomial (simplifies to \(0\))