(x^2-2x-8)(-x^2+3x-5)

Jasonbradley

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Jun 17, 2024

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Multiplying Polynomials: (x^2-2x-8)(-x^2+3x-5)

This article will guide you through the process of multiplying the two polynomials, (x^2-2x-8) and (-x^2+3x-5). We'll break down the steps and provide a clear explanation.

Understanding the Problem

We are given two trinomials:

• (x^2-2x-8)
• (-x^2+3x-5)

Our goal is to find the product of these two expressions.

The Multiplication Process

There are two common methods for multiplying polynomials:

• The Distributive Property: This method involves multiplying each term in the first polynomial by each term in the second polynomial.
• The FOIL Method: This is a shortcut for multiplying binomials, but it can be extended to multiplying trinomials.

Let's use the distributive property in this case:

1. Multiply the first term of the first polynomial (x^2) by each term of the second polynomial:

   • x^2 * (-x^2) = -x^4
   • x^2 * 3x = 3x^3
   • x^2 * (-5) = -5x^2

2. Multiply the second term of the first polynomial (-2x) by each term of the second polynomial:

   • -2x * (-x^2) = 2x^3
   • -2x * 3x = -6x^2
   • -2x * (-5) = 10x

3. Multiply the third term of the first polynomial (-8) by each term of the second polynomial:

   • -8 * (-x^2) = 8x^2
   • -8 * 3x = -24x
   • -8 * (-5) = 40

4. Combine all the terms:

   • -x^4 + 3x^3 - 5x^2 + 2x^3 - 6x^2 + 10x + 8x^2 - 24x + 40

5. Simplify by combining like terms:

   • -x^4 + 5x^3 - 3x^2 - 14x + 40

The Solution

Therefore, the product of the two polynomials (x^2-2x-8) and (-x^2+3x-5) is -x^4 + 5x^3 - 3x^2 - 14x + 40.