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: $(x^2+x+2)(x^2-2x+3)$.
Answer (1)
The degree of the product of two polynomials is the sum of their degrees, so the degree is 2 + 2 = 4 .
The maximum possible number of terms in the product is the product of the number of terms in each polynomial, which is 3 × 3 = 9 .
Multiplying the trinomials gives ( x 2 + x + 2 ) ( x 2 − 2 x + 3 ) = x 4 − x 3 + 3 x 2 − x + 6 .
The degree of the resulting polynomial is 4, and the simplified polynomial has 5 terms. The maximum possible number of terms before simplification is 9, so the final answers are degree 4 and maximum 9 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.
Determining the Degree To find the degree of the product, we add the degrees of the individual trinomials. Both trinomials have a degree of 2 (the highest power of x is x 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 multiply the number of terms in each trinomial. Each trinomial has 3 terms. Therefore, the maximum possible number of terms in the product is 3 × 3 = 9 . This occurs before combining any like terms.
Multiplying the Trinomials Now, let's multiply the two trinomials to find the actual number of terms and verify the degree: ( 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 ) = 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
Verifying the Results The resulting polynomial is x 4 − x 3 + 3 x 2 − x + 6 . The degree of this polynomial is 4, which matches our earlier calculation. The number of terms in the simplified polynomial is 5.
Final Answer Therefore, the degree of the product is 4, and the maximum possible number of terms is 9.
Examples
When designing a garden, you might want to calculate the area of a rectangular section where the length and width are described by polynomial expressions. Multiplying these polynomials helps you determine the total area as a function of a variable, like the number of plants or the size of a modular component. For example, if the length is ( x 2 + x + 2 ) meters and the width is ( x 2 − 2 x + 3 ) meters, the total area is ( x 2 + x + 2 ) ( x 2 − 2 x + 3 ) = x 4 − x 3 + 3 x 2 − x + 6 square meters. This allows you to plan the layout and resource allocation effectively.
