given a rectangular picture frame with the sides of 2x+1 and x^3+4, find the perimeter of the picture frame. Is it a polynomial? If so, find the degree of the polynomial, the leading coefficient, and the number of terms in the polynomial. If not, explain why.

To find the perimeter of a rectangular picture frame, we use the formula for the perimeter of a rectangle:

\[
P = 2(\text{length} + \text{width})
\]

### Step 1: Substituting the expressions for the sides
Given the sides of the picture frame as \( 2x + 1 \) and \( x^3 + 4 \), we substitute them into the perimeter formula:

\[
P = 2\left((2x + 1) + (x^3 + 4)\right)
\]

### Step 2: Simplifying the expression inside the parentheses
First, simplify the sum of the two side lengths:

\[
(2x + 1) + (x^3 + 4) = x^3 + 2x + (1 + 4) = x^3 + 2x + 5
\]

### Step 3: Multiplying by 2
Now, distribute the 2 to get the perimeter:

\[
P = 2(x^3 + 2x + 5) = 2x^3 + 4x + 10
\]

### Step 4: Answering the questions
- **Is it a polynomial?** Yes, the expression for the perimeter \( 2x^3 + 4x + 10 \) is a polynomial.
- **Degree of the polynomial:** The degree of the polynomial is 3, which is the highest power of \( x \).
- **Leading coefficient:** The leading coefficient is 2, which is the coefficient of the term with the highest degree (\( x^3 \)).
- **Number of terms:** The polynomial has 3 terms: \( 2x^3 \), \( 4x \), and \( 10 \).

Would you like any further details or clarifications?

Here are 5 related questions for practice:
1. What is the perimeter if \( x = 2 \)?
2. How would the perimeter change if one side length was doubled?
3. What would the area of the frame be?
4. How does the perimeter change if \( x \) increases by 1?
5. Can you identify the roots of the polynomial perimeter expression?

**Tip:** The degree of a polynomial indicates the highest power of the variable in the expression, and it tells you the general shape of the graph.

Given a rectangular picture frame with the sides of 2x+1 and x^3+4, find the perimeter of the picture frame. Is it a polynomial? If so, find the degree of the polynomial, the leading coefficient, and the number of terms in the polynomial. If not, explain why.

Learn how to calculate the perimeter of a rectangular picture frame with side lengths expressed as polynomials. This detailed explanation includes finding the polynomial perimeter, determining its degree, leading coefficient, and number of terms. Ideal for students in Grades 9-12.

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