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

Jasonbradley

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Jun 17, 2024

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Expanding and Simplifying the Expression (x⁴-2x³+2x-1)(x²-1)

This article will explore the process of expanding and simplifying the given expression: (x⁴-2x³+2x-1)(x²-1).

Expanding the Expression

We can expand the expression by using the distributive property (also known as the FOIL method). This means we multiply each term in the first polynomial by each term in the second polynomial.

Let's break it down step by step:

1. Multiply (x⁴) by each term in (x²-1):

   • x⁴ * x² = x⁶
   • x⁴ * -1 = -x⁴

2. Multiply (-2x³) by each term in (x²-1):

   • -2x³ * x² = -2x⁵
   • -2x³ * -1 = 2x³

3. Multiply (2x) by each term in (x²-1):

   • 2x * x² = 2x³
   • 2x * -1 = -2x

4. Multiply (-1) by each term in (x²-1):

   • -1 * x² = -x²
   • -1 * -1 = 1

Now, we have expanded the expression: (x⁴-2x³+2x-1)(x²-1) = x⁶ - x⁴ - 2x⁵ + 2x³ + 2x³ - 2x - x² + 1

Simplifying the Expression

To simplify the expression, we need to combine like terms:

• x⁶: There's only one term with x⁶, so it remains as is.
• x⁵: We have -2x⁵, so it remains as is.
• x⁴: We have -x⁴, so it remains as is.
• x³: We have 2x³ + 2x³, which simplifies to 4x³.
• x²: We have -x², so it remains as is.
• x: We have -2x, so it remains as is.
• Constant: We have 1, so it remains as is.

Therefore, the simplified expression is: (x⁴-2x³+2x-1)(x²-1) = x⁶ - 2x⁵ - x⁴ + 4x³ - x² - 2x + 1

Conclusion

By expanding and simplifying the expression (x⁴-2x³+2x-1)(x²-1), we obtain the polynomial x⁶ - 2x⁵ - x⁴ + 4x³ - x² - 2x + 1. This process demonstrates how to manipulate expressions involving polynomials and can be applied to various algebraic problems.