Question

if \\(f(-5) = 0\\), what are all the factors of the function \\(f(x) = x^3 - 19x + 30\\)? use the remainder theorem.

\\((x - 2)(x + 5)(x - 3)\\)
\\((x + 2)(x - 5)(x + 3)\\)
\\((x - 2)(x + 5)\\)
\\((x + 2)(x - 5)\\)

Explanation:

Identify the first factor

Using the Factor Theorem knowledge point

$$ f(-5) = 0 \implies (x + 5) \text{ is a factor of } f(x) $$

Perform polynomial division

Using the Polynomial Factoring knowledge point

$$ \frac{x^3 - 19x + 30}{x + 5} = x^2 - 5x + 6 $$

Factor the remaining quadratic

Using the Polynomial Factoring knowledge point

$$ x^2 - 5x + 6 = (x - 2)(x - 3) $$

Combine all factors

Using the Polynomial Factoring knowledge point

$$ f(x) = (x - 2)(x + 5)(x - 3) $$

Answer:

• (A) \((x-2)(x+5)(x-3)\) (Correct answer)
• (B) \((x+2)(x-5)(x+3)\)
• (C) \((x-2)(x+5)\)
• (D) \((x+2)(x-5)\)