Question

Answer the questions about the following polynomial.
1, minus, 7, x, cubed, plus, start fraction, x, divided by, 9, end fraction
1−7x
3
+
9
x
​
The expression represents a _____ polynomial with ____terms. The constant term is
____, the leading term is _____, and the leading coefficient is___.

quartic, cubic, linear, quadratic, quintic

Answer 1

To analyze the given polynomial \(1 - 7x^3 + \frac{x}{9}\):

1. **Identify the degree of the polynomial**: The highest power of \(x\) in the polynomial is \(3\) (from the term \(-7x^3\)), so this is a **cubic** polynomial.

2. **Count the terms**: The polynomial has three distinct terms: \(1\), \(-7x^3\), and \(\frac{x}{9}\). Therefore, it has **3 terms**.

3. **Identify the constant term**: The constant term is simply the term without \(x\), which is \(1\).

4. **Identify the leading term**: The leading term is the term with the highest power of \(x\), which is \(-7x^3\).

5. **Identify the leading coefficient**: The leading coefficient is the coefficient of the leading term. For the term \(-7x^3\), the leading coefficient is \(-7\).

Putting this all together, we have:

- The expression represents a **cubic** polynomial with **3** terms.
- The constant term is **1**.
- The leading term is **\(-7x^3\)**.
- The leading coefficient is **\(-7\)**.

So the filled answers are:

- **cubic** polynomial
- **3** terms
- Constant term is **1**
- Leading term is **\(-7x^3\)**
- Leading coefficient is **\(-7\)**.