Question

which second degree polynomial function \\(f(x)\\) has a lead coefficient of 3 and roots 4 and 1?

\\(f(x) = 3x^2 + 5x + 4\\)
\\(f(x) = 3x^2 + 15x + 12\\)
\\(f(x) = 3x^2 - 5x + 4\\)
\\(f(x) = 3x^2 - 15x + 12\\)

Explanation:

Write the factored form of the quadratic function

$$ f(x) = a(x - r_1)(x - r_2) $$

$$ a = 3,\quad r_1 = 4,\quad r_2 = 1 $$

$$ f(x) = 3(x - 4)(x - 1) $$

Expand the binomial product

$$ (x - 4)(x - 1) = x^2 - x - 4x + 4 = x^2 - 5x + 4 $$

Distribute the lead coefficient

$$ f(x) = 3(x^2 - 5x + 4) = 3x^2 - 15x + 12 $$

Answer:

• (A) \(f(x) = 3x^2 + 5x + 4\)
• (B) \(f(x) = 3x^2 + 15x + 12\)
• (C) \(f(x) = 3x^2 - 5x + 4\)
• (D) \(f(x) = 3x^2 - 15x + 12\) (Correct answer)