Alright, let's classify the given algebraic expressions into binomials and trinomials. Here's a detailed, step-by-step explanation:
### Definitions
1. Monomial: An algebraic expression with only one term (e.g., \(x^5\)).
2. Binomial: An algebraic expression with exactly two terms (e.g., \(x^2 + 3\)).
3. Trinomial: An algebraic expression with exactly three terms (e.g., \(x^3 - 2x - 3\)).
4. Other Polynomials: Algebraic expressions with more than three terms.
### Classifying the Expressions
Let's go through each given expression and count its terms.
#### 1. \(6x^2 + \frac{4x + 2}{2}\)
First, simplify:
[tex]\[ 6x^2 + \frac{4x + 2}{2} = 6x^2 + (2x + 1) \][/tex]
Simplified, the expression is:
[tex]\[ 6x^2 + 2x + 1 \][/tex]
This has three terms: \(6x^2\), \(2x\), and \(1\).
Classification: Trinomial
#### 2. \(\frac{5x + 2}{3}\)
This expression cannot be simplified further and it remains:
[tex]\[ \frac{5x + 2}{3} \][/tex]
This is considered as a single term.
Classification: Binomial
#### 3. \(x^3 - 2x - 3\)
The terms are: \(x^3\), \(-2x\), and \(-3\).
Classification: Trinomial
#### 4. \(x^3 + 9x - 3y + 6\)
The terms are: \(x^3\), \(9x\), \(-3y\), and \(6\).
Classification: Other Polynomial
#### 5. \(x^6 + 7x\)
The terms are: \(x^6\) and \(7x\).
Classification: Binomial
#### 6. \(x^5\)
This is a single term: \(x^5\).
Classification: Monomial
#### 7. \(y^4\)
This is a single term: \(y^4\).
Classification: Monomial
### Summary
Now let's summarize the classified expressions.
#### Binomials
- \(\frac{5x + 2}{3}\)
- \(x^6 + 7x\)
#### Trinomials
- \(6x^2 + \frac{4x + 2}{2}\) (or \(6x^2 + 2x + 1\) after simplification)
- \(x^3 - 2x - 3\)
#### Monomials
- \(x^5\)
- \(y^4\)
#### Other Polynomials
- \(x^3 + 9x - 3y + 6\)
### Final Result
- Binomials: \(\frac{5x + 2}{3}\), \(x^6 + 7x\)
- Trinomials: [tex]\(6x^2 + 2x + 1\)[/tex], [tex]\(x^3 - 2x - 3\)[/tex]
