Name the polynomial: 4x^5 + 6x^2 + 3 – Fifth-degree binomial – Fifth-degree monomial – Fifth-degree trinomial – Not a polynomial
The Correct Answer and Explanation is:
Correct Answer: Fifth-degree trinomial
The expression given is:
4x⁵ + 6x² + 3
To name a polynomial, we consider two things:
- Degree of the polynomial
- Number of terms in the expression
Step 1: Determine the Degree
The degree of a polynomial is the highest power of the variable (in this case, x) in the expression. Let’s look at each term:
- 4x⁵ has degree 5
- 6x² has degree 2
- 3 is a constant, so its degree is 0
The highest exponent is 5, so this is a fifth-degree polynomial.
Step 2: Count the Terms
A term is a part of a polynomial separated by a plus or minus sign.
The expression has:
- First term: 4x⁵
- Second term: 6x²
- Third term: 3
That makes three terms.
Step 3: Use the Correct Terminology
- A polynomial with one term is a monomial
- A polynomial with two terms is a binomial
- A polynomial with three terms is a trinomial
Since this expression has three terms, it is a trinomial.
Final Name
Putting both parts together:
- It is fifth-degree
- It is a trinomial
So, the correct classification is:
Fifth-degree trinomial
Why the Other Choices Are Incorrect
- Fifth-degree binomial: Incorrect because there are three terms, not two
- Fifth-degree monomial: Incorrect because there are three terms, not one
- Not a polynomial: Incorrect because the expression only contains whole number exponents and valid terms, which means it is indeed a polynomial
Therefore, the correct and complete name is fifth-degree trinomial.
