Question

which polynomial function has x-intercepts \\(-1\\), \\(0\\), and \\(2\\) and passes through the point \\((1, -6)\\)?

\\(f(x) = x^3 - x^2 - 2x\\)
\\(f(x) = 3x^3 - 3x^2 - 6x\\)
\\(f(x) = x^3 + x^2 - 2x\\)
\\(f(x) = 3x^3 + 3x^2 - 6x\\)

Explanation:

Set up the factored form of the polynomial

$$ f(x) = a(x - x_1)(x - x_2)(x - x_3) $$

$$ f(x) = ax(x + 1)(x - 2) $$

Solve for the leading coefficient

$$ f(1) = a(1)(1 + 1)(1 - 2) = -6 $$

$$ a(1)(2)(-1) = -6 \implies -2a = -6 \implies a = 3 $$

Expand the polynomial expression

$$ f(x) = 3x(x^2 - x - 2) $$

$$ f(x) = 3x^3 - 3x^2 - 6x $$

Answer:

• (A) \(f(x) = x^3 - x^2 - 2x\)
• (B) \(f(x) = 3x^3 - 3x^2 - 6x\) (Correct answer)
• (C) \(f(x) = x^3 + x^2 - 2x\)
• (D) \(f(x) = 3x^3 + 3x^2 - 6x\)