Sure, let's go through the process of determining the degree and the maximum possible number of terms for the product of two given trinomials, [tex]\((x^2 + x + 2)\)[/tex] and [tex]\((x^2 - 2x + 3)\)[/tex].
### 1. Determine the Degree of the Product
The degree of a polynomial is the highest power of the variable [tex]\(x\)[/tex] that appears in the polynomial.
- The first trinomial [tex]\((x^2 + x + 2)\)[/tex] has a degree of 2 because the highest power of [tex]\(x\)[/tex] is [tex]\(x^2\)[/tex].
- The second trinomial [tex]\((x^2 - 2x + 3)\)[/tex] also has a degree of 2 because the highest power of [tex]\(x\)[/tex] is [tex]\(x^2\)[/tex].
When you multiply two polynomials, the degree of the resulting polynomial is the sum of the degrees of the two polynomials:
[tex]\[
\text{Degree of the product} = \text{Degree of the first polynomial} + \text{Degree of the second polynomial}
\][/tex]
So,
[tex]\[
\text{Degree of the product} = 2 + 2 = 4
\][/tex]
### 2. Determine the Maximum Possible Number of Terms
Next, we determine the maximum possible number of terms in the product of two trinomials.
- Each trinomial has 3 terms.
When you multiply two polynomials, every term of the first polynomial will multiply with every term of the second polynomial. Therefore, the maximum number of terms in the product is the product of the number of terms in each polynomial:
[tex]\[
\text{Maximum number of terms} = (\text{Number of terms in the first polynomial}) \times (\text{Number of terms in the second polynomial})
\][/tex]
So,
[tex]\[
\text{Maximum number of terms} = 3 \times 3 = 9
\][/tex]
Thus, the degree of the product of [tex]\((x^2 + x + 2)\)[/tex] and [tex]\((x^2 - 2x + 3)\)[/tex] is 4, and the maximum possible number of terms is 9.
