-(x-2).jpeg "(x^3-3x^2+5x-6) (x-2)")
Multiplying Polynomials: (x^3 - 3x^2 + 5x - 6)(x - 2)
This article will explore the process of multiplying two polynomials: (x^3 - 3x^2 + 5x - 6) and (x - 2). We will utilize the distributive property to achieve this.
Understanding the Distributive Property
The distributive property states that multiplying a sum by a number is the same as multiplying each addend by the number and then adding the products. Symbolically:
a(b + c) = ab + ac
Applying the Distributive Property
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Distribute (x - 2) to each term in the first polynomial:
(x^3 - 3x^2 + 5x - 6)(x - 2) = x(x^3 - 3x^2 + 5x - 6) - 2(x^3 - 3x^2 + 5x - 6)
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Multiply each term inside the parentheses:
= (x^4 - 3x^3 + 5x^2 - 6x) + (-2x^3 + 6x^2 - 10x + 12)
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Combine like terms:
= x^4 - 3x^3 - 2x^3 + 5x^2 + 6x^2 - 6x - 10x + 12 = x^4 - 5x^3 + 11x^2 - 16x + 12
Conclusion
Therefore, the product of (x^3 - 3x^2 + 5x - 6) and (x - 2) is x^4 - 5x^3 + 11x^2 - 16x + 12. By understanding the distributive property and applying it systematically, we can effectively multiply polynomials of any degree.