Çözüldü Polynomial Division - Undetermined Coefficients Rule - Four Variable Equation

Determine if (3x - 4) is a factor of the polynomial (6x^4 - 17x^3 + 24x^2 - 34x + 24), show your solution.
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Solution - 1 (Polynomial Division)

https://i72.servimg.com/u/f72/19/97/10/39/polyno11.png

Since the remainder is zero, 3x - 4 is a factor of the polynomial 6x^4 - 17x^3 + 24x^2 - 34x + 24.

Another Solution Approach by Symbolab Calculator
https://www.symbolab.com/solver/polynomial-long-division-calculator/long division \frac{6x^{4}-17x^{3}+24x^{2}-34x+24}{3x-4}?or=input

Solution - 2 (Undetermined Coefficients Rule)
6x^4 - 17x^3 + 24x^2 - 34x + 24 = (3x - 4)(ax^3 + bx^2 + cx + d)
6x^4 - 17x^3 + 24x^2 - 34x + 24 = 3a·x^4 + (-4a + 3b)·x^3 + (-4b + 3c)·x^2 + (-4c + 3d)·x - 4d
Applying the rule;
6 = 3a ⇒ a = 2
-17 = -4a + 3b ⇒ b = (4a - 17) / 3 = (8 - 17) / 3 = -3
24 = -4b + 3c ⇒ c = (24 + 4b) / 3 = (24 - 12) / 3 = 4
24 = -4d ⇒ d = -6
In conclusion, 6x^4 - 17x^3 + 24x^2 - 34x + 24 = (3x - 4)(2x^3 - 3x^2 + 4x - 6), so 3x - 4 is a factor of the given polynomial.

Note:
Showing that 2x^3 - 3x^2 + 4x - 6 = (2x - 3)(x^2 + 2) is homework left for the interested students.

Solution - 3
3x - 4 = 0 ⇒ x = 4 / 3
P(x) = 6x^4 - 17x^3 + 24x^2 - 34x + 24
P(4 / 3) = 6·(4 / 3)^4 - 17·(4 / 3)^3 + 24·(4 / 3)^2 - 34·(4 / 3) + 24 = 0
The result 0 means that 3x - 4 is a factor of P(x).

Solution - 4
3x - 4 = x = 4 / 3....(I)
6x^4 - 17x^3 + 24x^2 - 34x + 24 = 0
According to the Rational Zero (Root) Theorem;
one of the factors of the coefficient of the lowest exponent (24x^0 = 24) is 4....(II)
one of the factors of the coefficient of the largest exponent (6) is 3....(III)
and since the ratio of (II) to (III) equals to (I), 3x - 4 is a factor of the given polynomial.