Question
Answer the questions about the following polynomial.
minus, 1, plus, 4, x, cubed, minus, start fraction, x, to the power 5 , divided by, 5, end fraction, plus, x, squared, minus, 3, x
−1+4x
3
−
5
x
5
+x
2
−3x
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
minus, 1, plus, 4, x, cubed, minus, start fraction, x, to the power 5 , divided by, 5, end fraction, plus, x, squared, minus, 3, x
−1+4x
3
−
5
x
5
+x
2
−3x
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
Let's analyze the given polynomial:
\[
-1 + 4x^3 - \frac{x^5}{5} + x^2 - 3x
\]
1. **Degree of the Polynomial**: The polynomial is defined by the term with the highest power of \(x\). In this case, the highest power is \(x^5\), which means this is a **quintic** polynomial (a polynomial of degree 5).
2. **Number of Terms**: To determine the number of terms, we look at each distinct part of the polynomial:
- \(-1\) (constant term)
- \(4x^3\)
- \(-\frac{x^5}{5}\)
- \(x^2\)
- \(-3x\)
This gives us a total of 5 terms.
3. **Constant Term**: The constant term in the polynomial is \(-1\).
4. **Leading Term**: The leading term is the term with the highest degree, which is \(-\frac{x^5}{5}\).
5. **Leading Coefficient**: The leading coefficient is the coefficient of the leading term, which is \(-\frac{1}{5}\).
Putting this all together:
The expression represents a **quintic** polynomial with **5 terms**. The constant term is **-1**, the leading term is **-\frac{x^5}{5}**, and the leading coefficient is **-\frac{1}{5}**.
\[
-1 + 4x^3 - \frac{x^5}{5} + x^2 - 3x
\]
1. **Degree of the Polynomial**: The polynomial is defined by the term with the highest power of \(x\). In this case, the highest power is \(x^5\), which means this is a **quintic** polynomial (a polynomial of degree 5).
2. **Number of Terms**: To determine the number of terms, we look at each distinct part of the polynomial:
- \(-1\) (constant term)
- \(4x^3\)
- \(-\frac{x^5}{5}\)
- \(x^2\)
- \(-3x\)
This gives us a total of 5 terms.
3. **Constant Term**: The constant term in the polynomial is \(-1\).
4. **Leading Term**: The leading term is the term with the highest degree, which is \(-\frac{x^5}{5}\).
5. **Leading Coefficient**: The leading coefficient is the coefficient of the leading term, which is \(-\frac{1}{5}\).
Putting this all together:
The expression represents a **quintic** polynomial with **5 terms**. The constant term is **-1**, the leading term is **-\frac{x^5}{5}**, and the leading coefficient is **-\frac{1}{5}**.