resultant polynomial is when we multiply given two polynomial but I always see that we sum up two polynomial in resultant polynomial.

The term "resultant polynomial" traditionally refers to a specific concept in algebra involving the elimination of one variable between two polynomials, often used to determine if they have a common root. However, in a more general context, when we combine two polynomials, we might be referring to their sum or product.

1. **Sum of Two Polynomials:**
   When we add two polynomials, we combine like terms. This is sometimes loosely referred to as their "resultant" in casual language, but it is more accurately the sum of the polynomials.

2. **Product of Two Polynomials:**
   When we multiply two polynomials, we perform a polynomial multiplication, which gives us the product.

Let's clarify the sum and product for the given polynomials:

1. **Sum of \(x^2 + x + 1\) and \(x^3 + x + 2x + 2\):**

   Simplify \(x^3 + x + 2x + 2\) to \(x^3 + 3x + 2\).

   Sum: \( (x^2 + x + 1) + (x^3 + 3x + 2) \)

   Combine like terms:

   \[ x^3 + x^2 + 4x + 3 \]

2. **Product of \(x^2 + x + 1\) and \(x^3 + x + 2x + 2\):**

   Simplify \(x^3 + x + 2x + 2\) to \(x^3 + 3x + 2\).

   Multiply: \((x^2 + x + 1) \times (x^3 + 3x + 2)\)

   Perform the multiplication:

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

   \[
   = x^5 + 3x^3 + 2x^2 + x^4 + 3x^2 + 2x + x^3 + 3x + 2
   \]

   Combine like terms:

   \[
   x^5 + x^4 + 4x^3 + 5x^2 + 5x + 2
   \]

If you have any questions or need further details, feel free to ask!

Here are some questions you might want to ask next:
1. How do you multiply polynomials step-by-step?
2. Can you show the steps to sum these specific polynomials together?
3. What are like terms in polynomials, and how do you combine them?
4. What is the degree of a polynomial sum?
5. What are common applications of polynomial sums and products?

**Tip:** When working with polynomials, always simplify your expression by combining like terms and ordering terms by their degree for clarity.

Explore the concept of resultant polynomials, focusing on the difference between sum and product operations. Learn how to add and multiply polynomials step-by-step with examples. Suitable for high school students studying algebra.

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