Question

What is the degree of 2x^4-3x^3+6 and x^3+5x^2-15. How did you determine the degree of the polynomials write a rule for finding the leading coefficient

Answer 1

The **degree** of a polynomial is defined as the highest power of the variable in that polynomial.

For the given polynomials:

1. **Polynomial: \(2x^4 - 3x^3 + 6\)**
- The highest power of \(x\) in this polynomial is 4 (from the term \(2x^4\)).
- Therefore, the **degree** of the polynomial \(2x^4 - 3x^3 + 6\) is **4**.

2. **Polynomial: \(x^3 + 5x^2 - 15\)**
- The highest power of \(x\) in this polynomial is 3 (from the term \(x^3\)).
- Therefore, the **degree** of the polynomial \(x^3 + 5x^2 - 15\) is **3**.

### Rule for Finding the Leading Coefficient
The **leading coefficient** of a polynomial is the coefficient of the term with the highest degree.

**To find the leading coefficient:**
1. Identify the term with the highest degree in the polynomial.
2. The coefficient of this term is the leading coefficient.

**Applying this rule:**
- For \(2x^4 - 3x^3 + 6\):
- The highest degree term is \(2x^4\).
- The leading coefficient is **2**.

- For \(x^3 + 5x^2 - 15\):
- The highest degree term is \(x^3\) (which has a coefficient of 1).
- The leading coefficient is **1**.

### Summary
- Degree of \(2x^4 - 3x^3 + 6\): 4, Leading coefficient: 2
- Degree of \(x^3 + 5x^2 - 15\): 3, Leading coefficient: 1