Certainly! To determine which expression is equivalent to [tex]\( f(x) = x^3 + 5x^2 + 3x - 9 \)[/tex] given that the polynomial intersects the x-axis at the points [tex]\((-3, 0)\)[/tex] and [tex]\((1, 0)\)[/tex], we'll derive the polynomial form using these roots.
### Step-by-Step Solution:
1. Identify the given roots:
- The polynomial [tex]\( f(x) \)[/tex] intersects the x-axis at [tex]\( x = -3 \)[/tex] and [tex]\( x = 1 \)[/tex].
- This means [tex]\((x + 3)\)[/tex] and [tex]\((x - 1)\)[/tex] are factors of [tex]\( f(x) \)[/tex].
2. General Form Using Roots:
- Since [tex]\( f(x) \)[/tex] is a cubic polynomial, it must be expressible in the form:
[tex]\[
f(x) = a(x + 3)(x - 1)(x - r)
\][/tex]
where [tex]\( r \)[/tex] is another root, and [tex]\( a \)[/tex] is a non-zero constant.
3. Expand and compare terms:
- We need to find out what the remaining factor is by polynomial division or by another method. For now, consider that the polynomial remains as shown with an unknown root and constant.
4. Use the constant term to solve for [tex]\(a\)[/tex]:
- Let's recognize that the product of all factors yields the constant term of [tex]\(-9\)[/tex], after evaluating the polynomial form fully.
- Find the constant term from the expanded form:
[tex]\[
a(x + 3)(x - 1)(x - r)
\][/tex]
5. Verification by Expanded Form:
- To check which expression matches [tex]\( x^3 + 5x^2 + 3x - 9 \)[/tex], we can look at options more closely.
- First, verify if there's a consistent term configuration matching [tex]\(f(x)\)[/tex]
### Analysis of Given Options:
Let's expand and verify which option matches the specified polynomial:
- Option A: [tex]\((x - 3)(x - 3)(x + 1)\)[/tex]
[tex]\[
(x - 3)(x - 3)(x + 1) = (x^2 - 6x + 9)(x + 1) = x^3 - 5x^2 + 3x + 9
\][/tex]
Not matching [tex]\( x^3 + 5x^2 + 3x - 9 \)[/tex]
- Option B: [tex]\((x - 3)(x + 1)(x + 1)\)[/tex]
[tex]\[
(x - 3)(x + 1)(x + 1) = (x^2 -2x-3)(x + 1) = x^3- x^2 -5x -3
\][/tex]
Not matching either
- Option C: [tex]\((x - 1)(x - 1)(x + 3)\)[/tex]
[tex]\[
(x - 1)(x - 1)(x + 3) = (x^2 - 2x + 1 )(x + 3) = x^3+x^2-5x -3, similarly to earlier differenc
- Option D: \((x - 1)(x + 3)(x + 3)\)
\[
(x - 1)(x + 3)(x + 3) = (x^2 + 6x + 9)(x - 1) = x^3+ 5x^2 + 3x- 9
\][/tex]
This finally matches up correctly with each polynomial term found.
From the expanded and correctly derived analysis, the Correct option is:
D [tex]\((x - 1)(x + 3)(x + 3)\)[/tex]
Thus, it satisfies that [tex]\( (x-1)(x + 3)(x + 3) \)[/tex] is indeed the expected expression matching to given [tex]\( f(x). \)[/tex]
