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___.
quartic, cubic, linear, quadratic, quintic
Answers
GPT-4o mini
The given polynomial is:
\[
\frac{1}{2}x^5 - 6 - x^4 - 5x^2
\]
1. **Degree**: The highest power of \(x\) in the polynomial is 5, which indicates that this is a **quintic polynomial**.
2. **Number of terms**: The polynomial consists of four terms: \(\frac{1}{2}x^5\), \(-x^4\), \(-5x^2\), and \(-6\). So, it has **4 terms**.
3. **Constant term**: The constant term is the term without \(x\), which is \(-6\).
4. **Leading term**: The leading term is the term with the highest degree, which is \(\frac{1}{2}x^5\).
5. **Leading coefficient**: The leading coefficient is the coefficient of the leading term, which is \(\frac{1}{2}\).
Putting it all together, the completed statement is:
The expression represents a **quintic** polynomial with **4** terms. The constant term is **-6**, the leading term is **\(\frac{1}{2}x^5\)**, and the leading coefficient is **\(\frac{1}{2}\)**.
\[
\frac{1}{2}x^5 - 6 - x^4 - 5x^2
\]
1. **Degree**: The highest power of \(x\) in the polynomial is 5, which indicates that this is a **quintic polynomial**.
2. **Number of terms**: The polynomial consists of four terms: \(\frac{1}{2}x^5\), \(-x^4\), \(-5x^2\), and \(-6\). So, it has **4 terms**.
3. **Constant term**: The constant term is the term without \(x\), which is \(-6\).
4. **Leading term**: The leading term is the term with the highest degree, which is \(\frac{1}{2}x^5\).
5. **Leading coefficient**: The leading coefficient is the coefficient of the leading term, which is \(\frac{1}{2}\).
Putting it all together, the completed statement is:
The expression represents a **quintic** polynomial with **4** terms. The constant term is **-6**, the leading term is **\(\frac{1}{2}x^5\)**, and the leading coefficient is **\(\frac{1}{2}\)**.