Question

the graph of the function f(x) = x³ + 5x² + 3x - 9 intersects the x-axis at the points (-3, 0) and (1, 0) as shown.
which expression is equivalent to x³ + 5x² + 3x - 9?
a (x - 3)(x - 3)(x + 1)
b (x - 3)(x + 1)(x + 1)
c (x - 1)(x - 1)(x + 3)
d (x - 1)(x + 3)(x + 3)

Explanation:

Step1: Recall Root-Factor Relationship

If a function \( f(x) \) has a root at \( x = a \), then \( (x - a) \) is a factor of \( f(x) \). From the graph and the problem, the roots are \( x = -3 \) (with multiplicity, since the graph touches or crosses? Wait, the roots are \( x = -3 \) and \( x = 1 \)? Wait, no, the function is cubic, so three roots (real or complex). The graph intersects the x-axis at \( (-3, 0) \) and \( (1, 0) \), but since it's a cubic, one root has multiplicity 2. Let's check the roots: if \( x = 1 \) is a root, then \( (x - 1) \) is a factor. If \( x = -3 \) is a root (with multiplicity 2, since the graph touches or has a turning point there), then \( (x + 3) \) is a factor (twice, because \( x = -3 \) implies \( x + 3 = 0 \), so factor is \( (x + 3) \)). Wait, let's test the options.

Step2: Test Each Option

Let's expand each option:

• Option A: \( (x - 3)(x - 3)(x + 1) \). Roots at \( x = 3 \) (double) and \( x = -1 \). Not matching given roots.
• Option B: \( (x - 3)(x + 1)(x + 1) \). Roots at \( x = 3 \) and \( x = -1 \) (double). Not matching.
• Option C: \( (x - 1)(x - 1)(x + 3) \). Roots at \( x = 1 \) (double) and \( x = -3 \). Let's expand: \( (x - 1)^2(x + 3) = (x^2 - 2x + 1)(x + 3) = x^3 + 3x^2 - 2x^2 - 6x + x + 3 = x^3 + x^2 - 5x + 3 \). Not matching the given polynomial \( x^3 + 5x^2 + 3x - 9 \).
• Option D: \( (x - 1)(x + 3)(x + 3) \). Expand: \( (x - 1)(x + 3)^2 = (x - 1)(x^2 + 6x + 9) = x^3 + 6x^2 + 9x - x^2 - 6x - 9 = x^3 + 5x^2 + 3x - 9 \). Which matches the given polynomial \( x^3 + 5x^2 + 3x - 9 \).

Answer:

D. \( (x - 1)(x + 3)(x + 3) \)