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: $(x^2+x+2)(x^2-2x+3)$. Explain how you arrived at your answer.

Multiply the two trinomials: ( x 2 + x + 2 ) ( x 2 − 2 x + 3 ) = x 4 − x 3 + 3 x 2 − x + 6 .
Identify the degree of the polynomial: The degree is 4.
Calculate the maximum possible number of terms before simplification: 3 × 3 = 9 .
State the degree and the maximum possible number of terms: The degree is 4, and the maximum possible number of terms is 9. The simplified expression has 5 terms, which is less than or equal to the maximum possible number of terms. 4 , 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.

Multiplying the Trinomials To find the degree and the maximum possible number of terms, we first need to multiply the two trinomials: ( x 2 + x + 2 ) ( x 2 − 2 x + 3 ) . Let's perform the multiplication:


( x 2 + x + 2 ) ( x 2 − 2 x + 3 ) = x 2 ( x 2 − 2 x + 3 ) + x ( x 2 − 2 x + 3 ) + 2 ( x 2 − 2 x + 3 )
= ( x 4 − 2 x 3 + 3 x 2 ) + ( x 3 − 2 x 2 + 3 x ) + ( 2 x 2 − 4 x + 6 )

Combining Like Terms Now, let's combine like terms:

x 4 + ( − 2 x 3 + x 3 ) + ( 3 x 2 − 2 x 2 + 2 x 2 ) + ( 3 x − 4 x ) + 6
= x 4 − x 3 + 3 x 2 − x + 6

Determining the Degree The degree of the resulting polynomial is the highest power of x , which is 4.

Determining the Maximum Possible Number of Terms To find the maximum possible number of terms, we assume that no terms combine during the multiplication process. In this case, each term in the first trinomial is multiplied by each term in the second trinomial. Since both trinomials have 3 terms, the maximum number of terms in the product is 3 × 3 = 9 . However, after simplification, we have 5 terms: x 4 , − x 3 , 3 x 2 , − x , and 6 .

Final Answer Therefore, the degree of the product is 4, and the maximum possible number of terms before simplification is 9. After simplification, the number of terms is 5.


Examples
Understanding polynomial multiplication is crucial in various fields like engineering, physics, and computer science. For instance, when designing a bridge, engineers use polynomial functions to model the load distribution. Multiplying these polynomials helps them predict the overall stress and ensure the bridge's stability. Similarly, in computer graphics, polynomial multiplication is used to create complex shapes and textures. By understanding the degree and number of terms, designers can optimize the rendering process and create visually appealing graphics efficiently. This concept also extends to financial modeling, where polynomial functions are used to predict market trends and manage investment risks.

Answered by GinnyAnswer | 2025-07-07

The degree of the product of the trinomials ( x 2 + x + 2 ) ( x 2 − 2 x + 3 ) is 4 , and the maximum possible number of terms before simplification is 9 . After combining like terms, the resulting polynomial has 5 terms. Therefore, the final results are 4 for the degree and 9 as the maximum possible number of terms.
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Answered by Anonymous | 2025-07-12