To address the question, first we will simplify the given polynomial by combining like terms and then analyze the resulting expression to determine its degree and its classification.
Given polynomial:
[tex]\[
\frac{1}{4} x^5 + \frac{1}{2} x^5 - 5 x - 11
\][/tex]
### Step 1: Combine Like Terms for [tex]\(x^5\)[/tex]
1. Identify the coefficients of [tex]\(x^5\)[/tex]:
[tex]\[
\frac{1}{4} x^5 + \frac{1}{2} x^5
\][/tex]
2. Find a common denominator for the fractions:
[tex]\[
\frac{1}{4} = \frac{1}{4} ; \quad \frac{1}{2} = \frac{2}{4}
\][/tex]
3. Combine the fractions:
[tex]\[
\frac{1}{4} x^5 + \frac{2}{4} x^5 = \left( \frac{1+2}{4} \right) x^5 = \frac{3}{4} x^5
\][/tex]
Therefore, the polynomial simplifies to:
[tex]\[
\frac{3}{4} x^5 - 5 x - 11
\][/tex]
### Step 2: Write in Descending Powers
The expression is already in descending powers:
[tex]\[
\frac{3}{4} x^5 - 5 x - 11
\][/tex]
### Step 3: Determine the Degree of the Polynomial
The degree of the polynomial is given by the highest power of [tex]\(x\)[/tex] in the polynomial. In this case, the highest power is [tex]\(x^5\)[/tex], so the degree is:
[tex]\[
5
\][/tex]
### Step 4: Classify the Polynomial
Classification is based on the number of terms in the polynomial:
- A monomial has 1 term
- A binomial has 2 terms
- A trinomial has 3 terms
This polynomial has 3 terms ([tex]\(\frac{3}{4} x^5\)[/tex], [tex]\(-5 x\)[/tex], and [tex]\(-11\)[/tex]), so it is a trinomial.
### Final Answer
B. The polynomial can be simplified.
[tex]\[
\frac{1}{4} x^5 + \frac{1}{2} x^5 - 5 x - 11 = \frac{3}{4} x^5 - 5 x - 11
\][/tex]
