Question

Answer the questions about the following polynomial.
one half, x, to the power 5 , minus, 6, minus, x, to the power 4 , minus, 5, x, squared
2
1
​
x
5
−6−x
4
−5x
2
The expression represents a _____ polynomial with ____terms. The constant term is
____, the leading term is _____, and the leading coefficient is___.

Answer 1

Let's analyze the given polynomial:

\[
\frac{1}{2}x^5 - 6 - x^4 - 5x^2
\]

1. **Degree of the polynomial:** The term with the highest power of \(x\) is \(\frac{1}{2}x^5\), which has a degree of 5. This means the polynomial is a **5th-degree** polynomial.

2. **Number of terms:** The polynomial can be rewritten, focusing on each term:

- \(\frac{1}{2}x^5\) (1st term)
- \(-x^4\) (2nd term)
- \(-5x^2\) (3rd term)
- \(-6\) (4th term)

This gives us a total of **4 terms**.

3. **Constant term:** The constant term is the term with no variable, which in this polynomial is \(-6\).

4. **Leading term:** The leading term (the term with the highest degree) is \(\frac{1}{2}x^5\).

5. **Leading coefficient:** The leading coefficient is the coefficient of the leading term, which in this case is \(\frac{1}{2}\).

Now, summarizing the information:

- The expression represents a **5th-degree** polynomial with **4 terms**.
- The constant term is **-6**, the leading term is **\(\frac{1}{2}x^5\)**, and the leading coefficient is **\(\frac{1}{2}\)**.