Question

Answer the questions about the following polynomial.
minus, 3, minus, x, plus, one third, x, squared, plus, x, to the power 4
−3−x+
3
1
​
x
2
+x
4

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

The given polynomial is:

\[
-x + \frac{1}{3} x^2 + x^4 - 3
\]

Let's rewrite it in standard form, arranging the terms from the highest degree to the lowest:

\[
x^4 + \frac{1}{3} x^2 - x - 3
\]

1. **Degree of the polynomial:** The highest power of \(x\) is 4, so this is a **quartic polynomial**.

2. **Number of terms:** The polynomial has four terms: \(x^4\), \(\frac{1}{3} x^2\), \(-x\), and \(-3\).

3. **Constant term:** The constant term is \(-3\).

4. **Leading term:** The leading term (the term with the highest degree) is \(x^4\).

5. **Leading coefficient:** The coefficient of the leading term \(x^4\) is 1.

Now, we can fill in the blanks:

- The expression represents a **quartic** polynomial with **4** terms.
- The constant term is **-3**, the leading term is **\(x^4\)**, and the leading coefficient is **1**.

So the final filled-out statement is:

The expression represents a quartic polynomial with 4 terms. The constant term is -3, the leading term is \(x^4\), and the leading coefficient is 1.