Question

give your answer in the form \\( q(x) + \frac{r(x)}{d(x)} \\)

Explanation:

🆕 New Concept Discovered: Synthetic Division
A shortcut method to divide polynomials by a linear factor.

Step 1: Understand the Synthetic Division Setup

The synthetic division setup shows:

• The divisor's root on the left: \( c = 4 \), which corresponds to dividing by \( x - 4 \).
• The coefficients of the dividend polynomial in the top row: \( 5 \), \( -8 \), \( 5 \), \( -2 \), \( 0 \), \( -5 \). This represents the polynomial:

$$ P(x) = 5x^5 - 8x^4 + 5x^3 - 2x^2 + 0x - 5 $$

Step 2: Complete the Synthetic Division Steps

We perform the synthetic division algorithm step-by-step:

1. Bring down the first coefficient:

• The first coefficient \( 5 \) is brought straight down to the bottom row.

1. First multiplication and addition:

• Multiply the root \( 4 \) by the bottom value \( 5 \): \( 4 \times 5 = 20 \) (this is already filled in the second row).
• Add the second column: \( -8 + 20 = 12 \).
• The value in the second box of the bottom row is \( 12 \) (currently showing a typo/placeholder \( 1 \) in the image).

1. Second multiplication and addition:

• Multiply the root \( 4 \) by the bottom value \( 12 \): \( 4 \times 12 = 48 \).
• Add the third column: \( 5 + 48 = 53 \).
• The bottom row value is \( 53 \).

1. Third multiplication and addition:

• Multiply the root \( 4 \) by the bottom value \( 53 \): \( 4 \times 53 = 212 \).
• Add the fourth column: \( -2 + 212 = 210 \).
• The bottom row value is \( 210 \).

1. Fourth multiplication and addition:

• Multiply the root \( 4 \) by the bottom value \( 210 \): \( 4 \times 210 = 840 \).
• Add the fifth column: \( 0 + 840 = 840 \).
• The bottom row value is \( 840 \).

1. Fifth multiplication and addition:

• Multiply the root \( 4 \) by the bottom value \( 840 \): \( 4 \times 840 = 3360 \).
• Add the sixth column: \( -5 + 3360 = 3355 \).
• The final remainder is \( 3355 \).

Step 3: Write the Quotient and Remainder

The bottom row coefficients (excluding the last remainder term) represent the quotient polynomial \( Q(x) \), which is one degree lower than the original polynomial:

$$ Q(x) = 5x^4 + 12x^3 + 53x^2 + 210x + 840 $$

The remainder is:

$$ R(x) = 3355 $$

The divisor is:

$$ D(x) = x - 4 $$

Now, we write the final expression in the form \( Q(x) + \frac{R(x)}{D(x)} \):

$$ 5x^4 + 12x^3 + 53x^2 + 210x + 840 + \frac{3355}{x-4} $$

Answer:

$$ 5x^4 + 12x^3 + 53x^2 + 210x + 840 + \frac{3355}{x-4} $$