By algebraic handling and factor rules, the polynomial (x - 2) · (x - 5) · (x - √3) · (x + √3) is equivalent to the expanded form x⁴ - 7 · x³ + 7 · x² + 21 · x - 30.
How to expand a quartic function
Herein we must transform a polynomial of the form (x - r₁) · (x - r₂) · (x - r₃) · (x - r₄) to the form a · x⁴ + b · x³ + c · x² + d · x + e by algebraic handling, especially factor rules:
(x - 2) · (x - 5) · (x - √3) · (x + √3)
(x² - 7 · x + 10) · (x² - 3) Factor rules
x² · (x² - 3) - (7 · x) · (x² - 3) + 10 · (x² - 3) Distributive and commutative properties
x⁴ - 3 · x² - 7 · x³ + 21 · x + 10 · x² - 30 Distributive property/Definition of power
x⁴ - 7 · x³ + 7 · x² + 21 · x - 30 Distributive property/Definition of addition
By algebraic handling and factor rules, the polynomial (x - 2) · (x - 5) · (x - √3) · (x + √3) is equivalent to the expanded form x⁴ - 7 · x³ + 7 · x² + 21 · x - 30.
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