(-x^3+2x^4-4-x^2+7x) (x^2+x-1)

Jasonbradley

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Jun 16, 2024

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Multiplying Polynomials: (-x^3+2x^4-4-x^2+7x) (x^2+x-1)

This article will guide you through the process of multiplying the two polynomials: (-x^3+2x^4-4-x^2+7x) and (x^2+x-1).

Understanding the Process

Multiplying polynomials involves distributing each term of the first polynomial to every term of the second polynomial. This process can be visualized as a grid, with each term of the first polynomial forming a row and each term of the second polynomial forming a column. The product of each row and column element forms a cell in the grid.

Step-by-Step Solution

1. Organize the Terms:

   • Arrange the terms of both polynomials in descending order of their exponents.
     • (-x^3+2x^4-4-x^2+7x) becomes (2x^4 - x^3 - x^2 + 7x - 4)
     • (x^2+x-1) remains as (x^2 + x - 1)

2. Create the Multiplication Grid:

   	x^2	x	-1
   2x^4	2x^6	2x^5	-2x^4
   -x^3	-x^5	-x^4	x^3
   -x^2	-x^4	-x^3	x^2
   7x	7x^3	7x^2	-7x
   -4	-4x^2	-4x	4

3. Multiply Each Term:

   • Multiply each term in the first polynomial by each term in the second polynomial.
   • For example, the first cell in the grid is the product of 2x^4 and x^2, which results in 2x^6.

4. Combine Like Terms:

   • Identify terms with the same exponents and add their coefficients.

   • x^6 term: 2x^6

   • x^5 term: 2x^5 - x^5 = x^5

   • x^4 term: -2x^4 - x^4 - x^4 = -4x^4

   • x^3 term: x^3 + 7x^3 = 8x^3

   • x^2 term: x^2 - 4x^2 = -3x^2

   • x term: 7x^2 - 4x = 3x^2 - 4x

   • Constant term: 4

5. Final Result:

   • Combine the simplified terms: 2x^6 + x^5 - 4x^4 + 8x^3 - 3x^2 + 3x^2 - 4x + 4

   • The final product of the two polynomials is 2x^6 + x^5 - 4x^4 + 8x^3 - 4x + 4.

Conclusion

By systematically distributing each term and combining like terms, we have successfully multiplied the polynomials (-x^3+2x^4-4-x^2+7x) and (x^2+x-1) to arrive at the final product 2x^6 + x^5 - 4x^4 + 8x^3 - 4x + 4. This process can be applied to any polynomial multiplication problem.