Question

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

Explanation:

🆕 New Concept Discovered: Synthetic Division
A shorthand method to divide polynomials by linear factors.

Step 1: Identify the components from the synthetic division table

The synthetic division setup shows:

• The divisor root on the left: \( c = 4 \), which corresponds to a divisor of \( D(x) = x - 4 \).
• The coefficients of the dividend polynomial in the top row: \( -5, 5, -2, 0, -5 \), representing the polynomial:

$$ -5x^4 + 5x^3 - 2x^2 + 0x - 5 $$

• The bottom row contains the coefficients of the quotient polynomial \( Q(x) \) and the remainder \( R(x) \):
• Quotient coefficients: \( -5, -15, -62, -248 \)
• Remainder: \( -997 \)

Step 2: Write the quotient polynomial \( Q(x) \)

Since the original polynomial was of degree 4 and we divided by a linear term, the quotient \( Q(x) \) will be of degree 3:

$$ Q(x) = -5x^3 - 15x^2 - 62x - 248 $$

Step 3: Write the final expression in the form \( Q(x) + \frac{R(x)}{D(x)} \)

Substitute \( Q(x) \), \( R(x) = -997 \), and \( D(x) = x - 4 \) into the required form:

$$ -5x^3 - 15x^2 - 62x - 248 + \frac{-997}{x - 4} $$

This can also be written as:

$$ -5x^3 - 15x^2 - 62x - 248 - \frac{997}{x - 4} $$

Answer:

$$ -5x^3 - 15x^2 - 62x - 248 - \frac{997}{x - 4} $$