(2x^4+4x^3-3x^2-4x+1) (x^2-1)

3 min read Jun 16, 2024

Multiplying Polynomials: (2x^4+4x^3-3x^2-4x+1) (x^2-1)

This article will guide you through the process of multiplying the two polynomials: (2x^4+4x^3-3x^2-4x+1) and (x^2-1).

Understanding the Process

Multiplying polynomials involves distributing each term of one polynomial to every term of the other. This can be achieved using the FOIL method (First, Outer, Inner, Last) for binomials, or a more general method for larger polynomials.

Applying the Distributive Property

Distribute the first term of the second polynomial (x^2) to all terms in the first polynomial:

x^2 * (2x^4+4x^3-3x^2-4x+1) = 2x^6 + 4x^5 - 3x^4 - 4x^3 + x^2
Distribute the second term of the second polynomial (-1) to all terms in the first polynomial:

-1 * (2x^4+4x^3-3x^2-4x+1) = -2x^4 - 4x^3 + 3x^2 + 4x - 1
Combine the results of step 1 and step 2:

2x^6 + 4x^5 - 3x^4 - 4x^3 + x^2 - 2x^4 - 4x^3 + 3x^2 + 4x - 1
Combine like terms:

2x^6 + 4x^5 - 5x^4 - 8x^3 + 4x^2 + 4x - 1

Final Result

Therefore, the product of the two polynomials (2x^4+4x^3-3x^2-4x+1) and (x^2-1) is 2x^6 + 4x^5 - 5x^4 - 8x^3 + 4x^2 + 4x - 1.

Key Points

Distributive Property: The key to multiplying polynomials is to distribute each term of one polynomial to all terms of the other.
Combining Like Terms: After multiplying, simplify the expression by combining like terms.
Order: While the order of multiplying terms doesn't affect the final result, it's often easier to start with the term with the highest degree.

Remember, practicing multiplying polynomials will help you become more comfortable with the process and understand how to simplify complex expressions.