Question

use synthetic division to find ((x^{5} + 4x^{4} + 24x^{2} + 3) div (x + 5)).
write your answer in the form (q(x) + \frac{r}{d(x)}), where (q(x)) is a polynomial, (r) is an integer, and (d(x)) is a linear polynomial. simplify any fractions.

Explanation:

Step1: Identify root for divisor

Divisor is $x + 5$, so root is $-5$. Polynomial coefficients (include missing $x^3$ and $x$ terms): $1, 4, 0, 24, 0, 3$.

Step2: Set up synthetic division

Write root $-5$ left, coefficients top:
$-5$ | $1$   $4$   $0$   $24$   $0$   $3$

Step3: Bring down first coefficient

Bring down $1$:
$-5$ | $1$   $4$   $0$   $24$   $0$   $3$
$1$

Step4: Multiply and add repeatedly

$1×(-5) = -5$; $4 + (-5) = -1$
$-1×(-5) = 5$; $0 + 5 = 5$
$5×(-5) = -25$; $24 + (-25) = -1$
$-1×(-5) = 5$; $0 + 5 = 5$
$5×(-5) = -25$; $3 + (-25) = -22$

Step5: Form quotient and remainder

Quotient coefficients: $1, -1, 5, -1, 5$ → $x^4 - x^3 + 5x^2 - x + 5$. Remainder: $-22$.

Step6: Write final form

$q(x) + \frac{r}{d(x)} = x^4 - x^3 + 5x^2 - x + 5 + \frac{-22}{x + 5}$

Answer:

$x^4 - x^3 + 5x^2 - x + 5 + \frac{-22}{x + 5}$