(5x^3-4x^2+2x)(-x^4+3x^3-2x^2+x)

Jasonbradley

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Jun 16, 2024

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Multiplying Polynomials: (5x^3-4x^2+2x)(-x^4+3x^3-2x^2+x)

This article will walk through the process of multiplying the two polynomials: (5x^3-4x^2+2x)(-x^4+3x^3-2x^2+x).

Understanding Polynomial Multiplication

Multiplying polynomials involves distributing each term of one polynomial to every term in the other polynomial. This can be done using the FOIL method for binomials, or simply by distributing systematically for larger polynomials.

Step-by-Step Solution

1. Distribute the first term of the first polynomial (5x^3) to each term in the second polynomial:

   (5x^3)(-x^4) + (5x^3)(3x^3) + (5x^3)(-2x^2) + (5x^3)(x)

   This simplifies to: -5x^7 + 15x^6 - 10x^5 + 5x^4

2. Distribute the second term of the first polynomial (-4x^2) to each term in the second polynomial:

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

   This simplifies to: 4x^6 - 12x^5 + 8x^4 - 4x^3

3. Distribute the third term of the first polynomial (2x) to each term in the second polynomial:

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

   This simplifies to: -2x^5 + 6x^4 - 4x^3 + 2x^2

4. Combine all the simplified terms:

   -5x^7 + 15x^6 - 10x^5 + 5x^4 + 4x^6 - 12x^5 + 8x^4 - 4x^3 - 2x^5 + 6x^4 - 4x^3 + 2x^2

5. Combine like terms:

   -5x^7 + 19x^6 - 26x^5 + 19x^4 - 8x^3 + 2x^2

Final Result

The product of the two polynomials is: -5x^7 + 19x^6 - 26x^5 + 19x^4 - 8x^3 + 2x^2.