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If f(–5) = 0, what are all the factors of the function f(x) = x^3 - 19x + 30. Use the Remainder Theorem

A (x – 2)(x + 5)(x – 3)
B (x + 2)(x – 5)(x + 3)
C (x – 2)(x + 5)
D (x + 2)(x – 5

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Hagrid
The right answer for the question that is being asked and shown above is that: "A (x – 2)(x + 5)(x – 3)." If f(–5) = 0, the factors of the function f(x) = x^3 - 19x + 30 is A (x – 2)(x + 5)(x – 3), using the Remainder Theorem
frika

If f(-5)=0, then x+5 is first factor of given polynomial.

1. Divide polynomial [tex] x^3 - 19x + 30 [/tex] by x+5:

[tex] x^3 - 19x + 30=(x+5)(x^2-5x+6) [/tex].

2. Factor the trinomial [tex] x^2-5x+6 [/tex]:

[tex] D=(-5)^2-4\cdot 6=25-24=1,\ \sqrt{D}=1,\\ \\ x_1=\dfrac{5-1}{2}=2, \ x_2=\dfrac{5+1}{2}=3,\\ \\
x^2-5x+6=(x-2)(x-3) [/tex].

3. Then

[tex] x^3 - 19x + 30=(x+5)(x-2)(x-3) [/tex].

Answer: correct choice is A.

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