Question

which of the numbers below are some potential roots of (p(x) = x^3 + 6x^2 - 7x - 60) according to the rational root theorem?

-10
-7
-5
3
15
24

use synthetic division to test one potential root. enter the numbers that complete the division problem.

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$$\begin{array}{r|rrrr} -5 & 1 & 6 & -7 & -60 \\\\ & & a & c & 60 \\\\ \\hline & 1 & b & d & 0 \\end{array}$$

\\

(a = \\)
(b = \\)
(c = \\)
(d = \\)

Explanation:

🆕 New Concept Discovered: Rational Root Theorem & Synthetic Division
Finding possible rational roots and testing them using synthetic division.

Step 1: Find potential roots using the Rational Root Theorem

According to the Rational Root Theorem, any potential rational root of the polynomial \( p(x) = a_n x^n + \dots + a_0 \) must be of the form \( \frac{p}{q} \), where:

• \( p \) is a factor of the constant term \( a_0 = -60 \)
• \( q \) is a factor of the leading coefficient \( a_n = 1 \)

Since the leading coefficient is \( 1 \), the potential rational roots are simply the integer factors of \( -60 \).

Let's list the factors of \( 60 \):
\[ \pm 1, \pm 2, \pm 3, \pm 4, \pm 5, \pm 6, \pm 10, \pm 12, \pm 15, \pm 20, \pm 30, \pm 60 \]

Now we check which of the given options are in this list:

• \( -10 \): Yes (factor of \( -60 \))
• \( -7 \): No
• \( -5 \): Yes (factor of \( -60 \))
• \( 3 \): Yes (factor of \( -60 \))
• \( 15 \): Yes (factor of \( -60 \))
• \( 24 \): No

Step 2: Complete the synthetic division

We are testing the potential root \( -5 \) using synthetic division for the polynomial \( p(x) = 1x^3 + 6x^2 - 7x - 60 \).

The setup is:

 -5 |   1    6   -7   -60
    |        a    c    60
    ---------------------
        1    b    d     0

Let's perform the steps of synthetic division:

1. Bring down the first coefficient: \( 1 \).
2. Multiply \( -5 \) by \( 1 \) to get \( a \):

\[ a = -5 \times 1 = -5 \]

1. Add the second column to get \( b \):

\[ b = 6 + a = 6 + (-5) = 1 \]

1. Multiply \( -5 \) by \( b \) to get \( c \):

\[ c = -5 \times 1 = -5 \]

1. Add the third column to get \( d \):

\[ d = -7 + c = -7 + (-5) = -12 \]

1. Multiply \( -5 \) by \( d \) to verify the next value:

\[ -5 \times (-12) = 60 \] (This matches the given value in the table)

1. Add the last column:

\[ -60 + 60 = 0 \]

Thus, the missing values are:

• \( a = -5 \)
• \( b = 1 \)
• \( c = -5 \)
• \( d = -12 \)

Answer:

Part 1: Potential Roots

The potential roots of \( p(x) \) from the given list are:

• \( -10 \)
• \( -5 \)
• \( 3 \)
• \( 15 \)

Part 2: Synthetic Division Values

• \( a = -5 \)
• \( b = 1 \)
• \( c = -5 \)
• \( d = -12 \)