-(2x%5E2-x%2B1).jpeg "(6x^3-5x^2+4x-1) (2x^2-x+1)")
Multiplying Polynomials: (6x^3 - 5x^2 + 4x - 1)(2x^2 - x + 1)
This article will guide you through the process of multiplying two polynomials: (6x^3 - 5x^2 + 4x - 1) and (2x^2 - x + 1).
Understanding the Process
Multiplying polynomials involves distributing each term of the first polynomial to every term of the second polynomial. We can visualize this as a table:
| 2x² | -x | 1 | |
|---|---|---|---|
| 6x³ | 12x⁵ | -6x⁴ | 6x³ |
| -5x² | -10x⁴ | 5x³ | -5x² |
| 4x | 8x³ | -4x² | 4x |
| -1 | -2x² | x | -1 |
Steps for Multiplication
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Distribute: Multiply each term of the first polynomial by each term of the second polynomial.
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Combine Like Terms: After distribution, combine terms with the same variable and exponent.
Calculating the Result
Following the steps above, we get:
- 12x⁵ + (-6x⁴ - 10x⁴) + (6x³ + 5x³ + 8x³) + (-5x² - 4x² - 2x²) + (4x + x) - 1
Simplifying:
12x⁵ - 16x⁴ + 19x³ - 11x² + 5x - 1
Conclusion
The product of (6x^3 - 5x^2 + 4x - 1) and (2x^2 - x + 1) is 12x⁵ - 16x⁴ + 19x³ - 11x² + 5x - 1. This process demonstrates the distributive property of multiplication and its application to polynomial expressions.