classify the following polynomials. combine any like terms first.
(2x - x^2 + 4x + x^2)
(x^3 - 5x^2 + 3 - 2)
(x^3 - x^3 + x^2 - x^2 + 3)
(7x - x^2 + x^3 + 4x^2)
(-6x - x^3 + 6x - 4x)

Question

Accepted Answer

# Explanation:

🆕 New Concept Discovered: Classifying Polynomials
Naming polynomials by degree and number of terms.

## Step 1: Understand how to classify polynomials

To classify a polynomial, we first simplify it by combining any like terms. Then, we name it based on two characteristics:
1. **Degree** (the highest exponent of the variable):
   - Degree 1: **linear**
   - Degree 2: **quadratic**
   - Degree 3: **cubic**
2. **Number of terms** (after simplifying):
   - 1 term: **monomial**
   - 2 terms: **binomial**
   - 3 terms: **trinomial**

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## Step 2: Classify $ 2x - x^2 + 4x + x^2 $

Combine like terms:
- Combine $ x^2 $ terms: $ -x^2 + x^2 = 0 $
- Combine $ x $ terms: $ 2x + 4x = 6x $

Simplified expression:
$$ 6x $$

- Degree: 1 (linear)
- Number of terms: 1 (monomial)

**Classification:** linear monomial

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## Step 3: Classify $ x^3 - 5x^2 + 3 - 2 $

Combine like terms:
- Combine constant terms: $ 3 - 2 = 1 $

Simplified expression:
$$ x^3 - 5x^2 + 1 $$

- Degree: 3 (cubic)
- Number of terms: 3 (trinomial)

**Classification:** cubic trinomial

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## Step 4: Classify $ x^3 - x^3 + x^2 - x^2 + 3 $

Combine like terms:
- Combine $ x^3 $ terms: $ x^3 - x^3 = 0 $
- Combine $ x^2 $ terms: $ x^2 - x^2 = 0 $

Simplified expression:
$$ 3 $$

- Degree: 0 (constant)
- Number of terms: 1 (monomial)

**Classification:** constant monomial

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## Step 5: Classify $ 7x - x^2 + x^2 + 4x^2 $

Combine like terms:
- Combine $ x^2 $ terms: $ -x^2 + x^2 + 4x^2 = 4x^2 $

Simplified expression:
$$ 4x^2 + 7x $$

- Degree: 2 (quadratic)
- Number of terms: 2 (binomial)

**Classification:** quadratic binomial

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## Step 6: Classify $ -6x - x^3 + 6x - 4x $

Combine like terms:
- Combine $ x $ terms: $ -6x + 6x - 4x = -4x $

Simplified expression:
$$ -x^3 - 4x $$

- Degree: 3 (cubic)
- Number of terms: 2 (binomial)

**Classification:** cubic binomial

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# Answer:

* **$ 2x - x^2 + 4x + x^2 $** $
ightarrow$ **linear monomial**
* **$ x^3 - 5x^2 + 3 - 2 $** $
ightarrow$ **cubic trinomial**
* **$ x^3 - x^3 + x^2 - x^2 + 3 $** $
ightarrow$ **constant monomial**
* **$ 7x - x^2 + x^2 + 4x^2 $** $
ightarrow$ **quadratic binomial**
* **$ -6x - x^3 + 6x - 4x $** $
ightarrow$ **cubic binomial**

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Step 1: Understand how to classify polynomials

Step 2: Classify \( 2x - x^2 + 4x + x^2 \)

Step 3: Classify \( x^3 - 5x^2 + 3 - 2 \)

Step 4: Classify \( x^3 - x^3 + x^2 - x^2 + 3 \)

Step 5: Classify \( 7x - x^2 + x^2 + 4x^2 \)

Step 6: Classify \( -6x - x^3 + 6x - 4x \)

Answer: