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

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)

Multiply each term inside the parentheses:
= (x^4  3x^3 + 5x^2  6x) + (2x^3 + 6x^2  10x + 12)

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.