Expanding the Expression: (x^3 + 3x^2 + 1)(3x^2 + 6x  2)
This article explores the process of expanding the given polynomial expression: (x^3 + 3x^2 + 1)(3x^2 + 6x  2).
Understanding the Concept
The expression involves the multiplication of two polynomials. To expand it, we can use the distributive property, also known as the FOIL method for binomials. The FOIL method stands for First, Outer, Inner, Last, which refers to the terms we multiply in each step.
Expanding the Expression

First: Multiply the first term of each polynomial: (x^3) * (3x^2) = 3x^5

Outer: Multiply the outer terms of the polynomials: (x^3) * (6x) = 6x^4

Inner: Multiply the inner terms of the polynomials: (3x^2) * (3x^2) = 9x^4

Last: Multiply the last terms of each polynomial: (3x^2) * (2) = 6x^2

Repeat for remaining terms: We repeat the FOIL process for the remaining terms of the first polynomial (1) and multiply it with each term of the second polynomial (3x^2 + 6x  2).

Combine like terms: After expanding the entire expression, we combine terms with the same powers of x.
Resulting Polynomial
After performing all the multiplications and combining like terms, we obtain the following expanded polynomial:
(x^3 + 3x^2 + 1)(3x^2 + 6x  2) = 3x^5 + 15x^4 + 12x^3  6x^2 + 6x  2
Summary
By using the distributive property (FOIL method), we successfully expanded the given polynomial expression. The final result is a polynomial of degree 5 with six terms. This process can be applied to expand any expression involving the multiplication of polynomials.