To classify the expression [tex]\( 5x + 3x^4 - 7x^3 + 10 \)[/tex], we need to determine two key pieces of information: the number of terms and the degree of the polynomial.
### Step-by-Step Solution
1. Identify the Number of Terms:
- The given expression is [tex]\( 5x + 3x^4 - 7x^3 + 10 \)[/tex].
- We can break it down into the individual terms: [tex]\( 5x \)[/tex], [tex]\( 3x^4 \)[/tex], [tex]\( -7x^3 \)[/tex], and [tex]\( 10 \)[/tex].
- Therefore, the expression consists of 4 distinct terms.
2. Determine the Degree of the Polynomial:
- The degree of a polynomial is determined by the term with the highest power of [tex]\( x \)[/tex].
- In the expression [tex]\( 5x + 3x^4 - 7x^3 + 10 \)[/tex], the powers of [tex]\( x \)[/tex] for each term are as follows:
- [tex]\( 5x \)[/tex] has a power of 1.
- [tex]\( 3x^4 \)[/tex] has a power of 4.
- [tex]\( -7x^3 \)[/tex] has a power of 3.
- [tex]\( 10 \)[/tex] (constant term) has a power of 0 because it can be written as [tex]\( 10 \cdot x^0 \)[/tex].
- The highest power among these is 4.
3. Classification:
- Since the number of terms is 4, and the highest degree is 4, we classify this polynomial based on its degree and number of terms.
- A polynomial of degree 4 with any number of terms is generally referred to as a "fourth degree polynomial."
Thus, the expression [tex]\( 5x + 3x^4 - 7x^3 + 10 \)[/tex] is classified as a "fourth degree polynomial."
### Final Answer:
- The correct classification of the polynomial [tex]\( 5x + 3x^4 - 7x^3 + 10 \)[/tex] is fourth degree polynomial.
