Let's solve the given polynomial [tex]\(5x + 3x^5 - 7x^3 + 10\)[/tex] step by step:
### Part a: How many terms are there?
A polynomial term is a product of a coefficient and a variable raised to a power. To find the number of terms, we count each distinct term in the expression.
The given polynomial is:
[tex]\[ 5x + 3x^5 - 7x^3 + 10 \][/tex]
We identify four terms in the expression:
1. [tex]\(5x\)[/tex]
2. [tex]\(3x^5\)[/tex]
3. [tex]\(-7x^3\)[/tex]
4. [tex]\(10\)[/tex]
So, there are 4 terms in the polynomial.
### Part b: Put the expression in standard form. How do you know it's form?
The standard form of a polynomial arranges the terms in descending order based on the exponent of the variable.
Given polynomial:
[tex]\[ 5x + 3x^5 - 7x^3 + 10 \][/tex]
To write this in standard form:
1. Identify the degree of each term:
- [tex]\(3x^5\)[/tex] (degree 5)
- [tex]\(-7x^3\)[/tex] (degree 3)
- [tex]\(5x\)[/tex] (degree 1)
- [tex]\(10\)[/tex] (degree 0)
2. Rearrange the terms in descending order of their degrees:
- Degree 5: [tex]\(3x^5\)[/tex]
- Degree 3: [tex]\(-7x^3\)[/tex]
- Degree 1: [tex]\(5x\)[/tex]
- Constant term (degree 0): [tex]\(10\)[/tex]
So, the polynomial in standard form is:
[tex]\[ 3x^5 - 7x^3 + 5x + 10 \][/tex]
### Part c: What is the degree of the polynomial?
The degree of a polynomial is the highest degree (or exponent) of the variable in any of its terms.
From the standard form we obtained:
[tex]\[ 3x^5 - 7x^3 + 5x + 10 \][/tex]
We evaluate the degrees:
- [tex]\(3x^5\)[/tex] has a degree of 5
- [tex]\(-7x^3\)[/tex] has a degree of 3
- [tex]\(5x\)[/tex] has a degree of 1
- [tex]\(10\)[/tex] has a degree of 0
The highest degree among these is 5.
Therefore, the degree of the polynomial is 5.
