(x^3-8)(x^2-2x+1)-(3x^4-4x^2+16x-4)

2 min read Jun 17, 2024

Simplifying the Expression: (x³-8)(x²-2x+1)-(3x⁴-4x²+16x-4)

This article will guide you through simplifying the given algebraic expression: (x³-8)(x²-2x+1)-(3x⁴-4x²+16x-4). We'll break down each step and use algebraic properties to achieve a simplified form.

Step 1: Expanding the Products

First, we need to expand the products in the expression.

(x³-8)(x²-2x+1): This is a product of two binomials. We can use the distributive property (or FOIL method) to multiply them:

(x³)(x²) + (x³)(-2x) + (x³)(1) + (-8)(x²) + (-8)(-2x) + (-8)(1) = x⁵ - 2x⁴ + x³ - 8x² + 16x - 8
(3x⁴-4x²+16x-4): This expression is already expanded.

Step 2: Combining the Expanded Terms

Now, let's combine the expanded terms:

(x⁵ - 2x⁴ + x³ - 8x² + 16x - 8) - (3x⁴ - 4x² + 16x - 4)

Since we are subtracting the entire second expression, we need to distribute the negative sign:

x⁵ - 2x⁴ + x³ - 8x² + 16x - 8 - 3x⁴ + 4x² - 16x + 4

Step 3: Combining Like Terms

Finally, combine like terms:

x⁵ + (-2x⁴ - 3x⁴) + x³ + (-8x² + 4x²) + (16x - 16x) + (-8 + 4)

Simplifying:

x⁵ - 5x⁴ + x³ - 4x² - 4

Conclusion

The simplified form of the expression (x³-8)(x²-2x+1)-(3x⁴-4x²+16x-4) is x⁵ - 5x⁴ + x³ - 4x² - 4.