Question

divide \\(-5x^4 + 5x^3 - 2x^2 - 5\\) by \\(x - 4\\)

use the table below to complete the synthetic division.

\

$$\begin{array}{r|rrrrr} 4 & -5 & 5 & -2 & 0 & -5 \\\\ & & 20 & & & \\\\ \\hline & 5 & & & & \\end{array}$$

Explanation:

🆕 New Concept Discovered: Synthetic Division
A shorthand method of dividing polynomials using only coefficients.

Step 1: Identify the setup and first column

The division is of \( -5x^4 + 5x^3 - 2x^2 - 5 \) by \( x - 4 \).

The divisor is \( x - 4 \), so the test value on the left is \( 4 \).
The coefficients of the dividend are:

• For \( x^4 \): \( -5 \)
• For \( x^3 \): \( 5 \)
• For \( x^2 \): \( -2 \)
• For \( x^1 \): \( 0 \) (placeholder for the missing \( x \) term)
• Constant: \( -5 \)

The first coefficient \( -5 \) is brought straight down to the bottom row:

$$ \text{Bottom row, column 1} = -5 $$

(Note: The image shows a typo in the first box of the bottom row as "5", but it should be "-5" based on standard synthetic division. We will proceed with the correct mathematical steps using the brought-down value of \( -5 \).)

Step 2: Calculate the second column

Multiply the divisor value \( 4 \) by the bottom value \( -5 \):

$$ 4 \times (-5) = -20 $$

(Note: The image shows "20" in the second row, which is a common sign error in the template, or it assumes subtraction. However, standard synthetic division uses addition. Let's follow the standard addition-based synthetic division where we multiply \( 4 \times (-5) = -20 \), and then add \( 5 + (-20) = -15 \). If the template has a fixed "+20", then \( 5 + 20 = 25 \). Looking closely at the blue highlighted box, it has a "1" entered, which suggests the user might be testing inputs. Let's complete the standard synthetic division table step-by-step.)

Using standard synthetic division (adding columns):

1. First column:

• Top: \( -5 \)
• Bottom: \( -5 \)

1. Second column:

• Top: \( 5 \)
• Middle: \( 4 \times (-5) = -20 \)
• Bottom: \( 5 + (-20) = -15 \)

1. Third column:

• Top: \( -2 \)
• Middle: \( 4 \times (-15) = -60 \)
• Bottom: \( -2 + (-60) = -62 \)

1. Fourth column:

• Top: \( 0 \)
• Middle: \( 4 \times (-62) = -248 \)
• Bottom: \( 0 + (-248) = -248 \)

1. Fifth column:

• Top: \( -5 \)
• Middle: \( 4 \times (-248) = -992 \)
• Bottom: \( -5 + (-992) = -997 \)

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Alternative Interpretation (If the platform uses subtraction-based division)

If the platform's template has a positive \( 20 \) in the second column and expects subtraction:

• Column 1 bottom: \( -5 \)
• Column 2 middle: \( 20 \) (which is \( 4 \times -5 = -20 \), but written as positive if subtracting: \( 5 - (-20) = 25 \))
• Since the user entered \( 1 \) in the second column bottom, let's look at the relation: if the middle is \( 20 \) and the bottom is \( 1 \), it does not align with standard math.

Assuming standard synthetic division with addition (and correcting the typo in the first bottom box to \( -5 \)):

• Middle row boxes from left to right: [ -20 ], [ -60 ], [ -248 ], [ -992 ]
• Bottom row boxes from left to right: [ -5 ], [ -15 ], [ -62 ], [ -248 ], [ -997 ]

Answer:

The completed standard synthetic division table is:

• Middle row (from left to right, starting at column 2):

$$ -20,\quad -60,\quad -248,\quad -992 $$

• Bottom row (from left to right):

$$ -5,\quad -15,\quad -62,\quad -248,\quad -997 $$