To determine the degree and the maximum possible number of terms in the product of the trinomials [tex]\((x^2 + x + 2)\)[/tex] and [tex]\((x^2 - 2x + 3)\)[/tex], let's break down the multiplication process and analyze it step by step.
### Step 1: Understand the Degree of the Product Polynomial
First, consider the degrees of the given trinomials:
- The degree of [tex]\((x^2 + x + 2)\)[/tex] is 2, since the highest power of [tex]\(x\)[/tex] is [tex]\(x^2\)[/tex].
- The degree of [tex]\((x^2 - 2x + 3)\)[/tex] is also 2, as the highest power of [tex]\(x\)[/tex] is [tex]\(x^2\)[/tex].
When multiplying two polynomials, the degree of the resulting polynomial is the sum of the degrees of the original polynomials:
[tex]\[ \text{Degree of product} = 2 + 2 = 4 \][/tex]
### Step 2: Maximum Possible Number of Terms in the Product Polynomial
Next, let's determine the maximum possible number of terms in the product polynomial. Each term in the first polynomial has the potential to multiply with every term in the second polynomial.
Here are the terms in each trinomial:
- [tex]\((x^2 + x + 2)\)[/tex] has 3 terms: [tex]\(x^2\)[/tex], [tex]\(x\)[/tex], and [tex]\(2\)[/tex].
- [tex]\((x^2 - 2x + 3)\)[/tex] also has 3 terms: [tex]\(x^2\)[/tex], [tex]\(-2x\)[/tex], and [tex]\(3\)[/tex].
When we multiply each term from the first polynomial by each term from the second polynomial, the total number of resulting terms will be:
[tex]\[ \text{Number of terms in the product} = 3 \times 3 = 9 \][/tex]
### Conclusion
1. Degree of the Product Polynomial: The degree of the product polynomial [tex]\((x^2 + x + 2)(x^2 - 2x + 3)\)[/tex] is 4.
2. Maximum Possible Number of Terms: The maximum possible number of terms in the product is 9.
So, the degree of the product polynomial is 4, and the maximum possible number of terms in the product polynomial is 9.
