Multiply.
$\left(x^4+1\right)\left(3 x^2+9 x+2\right)$
Answer (2)
Distribute x 4 and 1 over the terms of the second polynomial.
Expand each term: x 4 ( 3 x 2 + 9 x + 2 ) = 3 x 6 + 9 x 5 + 2 x 4 and 1 ( 3 x 2 + 9 x + 2 ) = 3 x 2 + 9 x + 2 .
Combine the expanded terms: 3 x 6 + 9 x 5 + 2 x 4 + 3 x 2 + 9 x + 2 .
The final result is: 3 x 6 + 9 x 5 + 2 x 4 + 3 x 2 + 9 x + 2 .
Explanation
Understanding the Problem We are given the expression ( x 4 + 1 ) ( 3 x 2 + 9 x + 2 ) to expand. Our goal is to multiply these two polynomials together and simplify the result.
Distributing the Terms To multiply the polynomials, we distribute each term of the first polynomial ( x 4 + 1 ) over each term of the second polynomial ( 3 x 2 + 9 x + 2 ) . This can be written as: x 4 ( 3 x 2 + 9 x + 2 ) + 1 ( 3 x 2 + 9 x + 2 )
Expanding the Terms Now, we expand each term: x 4 ( 3 x 2 + 9 x + 2 ) = 3 x 6 + 9 x 5 + 2 x 4 1 ( 3 x 2 + 9 x + 2 ) = 3 x 2 + 9 x + 2 So, the expression becomes: 3 x 6 + 9 x 5 + 2 x 4 + 3 x 2 + 9 x + 2
Combining Like Terms Next, we combine like terms. In this case, there are no like terms to combine, as each term has a different power of x .
Final Result Finally, we write the polynomial in standard form, which means arranging the terms in descending order of the powers of x . The expanded polynomial is: 3 x 6 + 9 x 5 + 2 x 4 + 0 x 3 + 3 x 2 + 9 x + 2 So, the final result is: 3 x 6 + 9 x 5 + 2 x 4 + 3 x 2 + 9 x + 2
Examples
Polynomial multiplication is a fundamental concept in algebra and has many real-world applications. For example, engineers use polynomial multiplication to model the behavior of systems, such as the trajectory of a rocket or the stress on a bridge. In computer graphics, polynomial multiplication is used to perform transformations on objects, such as scaling, rotation, and translation. Understanding polynomial multiplication is essential for anyone working in these fields.
To multiply ( x 4 + 1 ) ( 3 x 2 + 9 x + 2 ) , we use the distributive property. After expanding and combining like terms, the final result is 3 x 6 + 9 x 5 + 2 x 4 + 3 x 2 + 9 x + 2 . This polynomial expresses the product of the two original polynomials in standard form.
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