Question

select all of the following that are potential roots of (p(x) = x^3 - 9x^2 - 4x + 12):

0
pm 2
pm 4
pm 9
\frac{1}{2}
pm 3
pm 6
pm 12

Explanation:

🆕 New Concept Discovered: Rational Root Theorem
Finding possible rational solutions of a polynomial equation.

Step 1: Identify the polynomial coefficients

The given polynomial function is:
\[ p(x) = x^3 - 9x^2 - 4x + 12 \]

Identify the leading coefficient \( a_n \) and the constant term \( a_0 \):

• Leading coefficient \( a_n = 1 \)
• Constant term \( a_0 = 12 \)

Step 2: Apply the Rational Root Theorem

The Rational Root Theorem states that any potential rational root of a polynomial must be of the form \( \pm \frac{p}{q} \), where:

• \( p \) is a factor of the constant term \( a_0 = 12 \)
• \( q \) is a factor of the leading coefficient \( a_n = 1 \)

Find the factors of the constant term \( 12 \):
\[ p \in \{ \pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12 \} \]

Find the factors of the leading coefficient \( 1 \):
\[ q \in \{ \pm 1 \} \]

Step 3: List all potential rational roots

Form the possible ratios \( \frac{p}{q} \):
\[ \pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12 \]

Now, match these potential roots with the options visible in the image:

• \( 9 \) (Not a factor of 12)
• \( -12 \) (Is a factor of 12)
• \( \pm 4 \) (Are factors of 12)
• \( \pm 9 \) (Not factors of 12)
• \( \frac{1}{3} \) (Not an integer factor of 12)
• \( \pm 3 \) (Are factors of 12)
• \( \pm 6 \) (Are factors of 12)
• \( \pm 12 \) (Are factors of 12)

From the given list, the values that are potential rational roots are:

• \( -12 \)
• \( \pm 4 \)
• \( \pm 3 \)
• \( \pm 6 \)
• \( \pm 12 \)

Answer:

The potential rational roots from the given options are:

• \( -12 \)
• \( \pm 4 \)
• \( \pm 3 \)
• \( \pm 6 \)
• \( \pm 12 \)