Multiplying a trinomial by a trinomial follows the same steps as multiplying a binomial by a trinomial. Determine the degree and maximum possible number of terms for the product of these trinomials: $\left(x^2+x+2\right)\left(x^2-2 x+3\right)$. Explain how you arrived at your answer.
Answer (1)
The degree of the product is the sum of the degrees of the trinomials: 2 + 2 = 4 .
The maximum number of terms is the product of the number of terms in each trinomial: 3 × 3 = 9 .
The degree of the product is 4 .
The maximum possible number of terms is 9 .
Explanation
Understanding the Problem We are given two trinomials: ( x 2 + x + 2 ) and ( x 2 − 2 x + 3 ) . We need to find the degree of their product and the maximum possible number of terms in their product.
Finding the Degree The degree of a polynomial is the highest power of the variable in the polynomial. To find the degree of the product of two polynomials, we add their degrees. The degree of ( x 2 + x + 2 ) is 2, and the degree of ( x 2 − 2 x + 3 ) is also 2. Therefore, the degree of the product is 2 + 2 = 4 .
Finding the Maximum Number of Terms To find the maximum possible number of terms in the product, we assume that no terms combine when we multiply the trinomials. When we multiply ( x 2 + x + 2 ) ( x 2 − 2 x + 3 ) , each term in the first trinomial is multiplied by each term in the second trinomial. This gives us 3 × 3 = 9 terms. Therefore, the maximum possible number of terms is 9.
Final Answer The degree of the product of the trinomials is 4, and the maximum possible number of terms is 9.
Examples
Understanding polynomial degrees and term multiplication is crucial in fields like computer graphics, where complex shapes are modeled using polynomial surfaces. Determining the degree helps optimize rendering algorithms, while knowing the maximum number of terms aids in memory allocation for storing shape data. For instance, calculating the lighting on a surface might involve multiplying several polynomial expressions, and efficient computation relies on knowing the degree and potential number of terms in the resulting polynomial.
