Question

use synthetic division to simplify \\(\frac{x^4 - 16x^3 - 40}{x - 1}\\).
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:

division
Write root & coefficients:
1 | 1 -16 0 0 -40

Step3: Bring down first coefficient

Bring down 1:
1 | 1 -16 0 0 -40
1
1

Step4: Multiply and add (x³ term)

$1×1=1$; $-16+1=-15$:
1 | 1 -16 0 0 -40
1 -15
1 -15

Step5: Multiply and add (x² term)

$1×(-15)=-15$; $0+(-15)=-15$:
1 | 1 -16 0 0 -40
1 -15 -15
1 -15 -15

Step6: Multiply and add (x term)

$1×(-15)=-15$; $0+(-15)=-15$:
1 | 1 -16 0 0 -40
1 -15 -15 -15
1 -15 -15 -15

Step7: Multiply and add (remainder)

$1×(-15)=-15$; $-40+(-15)=-55$:
1 | 1 -16 0 0 -40
1 -15 -15 -15
1 -15 -15 -15 -55

Step8: Form the result

Quotient: $x^3 -15x^2 -15x -15$, remainder: $-55$.
Result: $x^3 -15x^2 -15x -15 + \frac{-55}{x-1}$

Answer:

division
Write root & coefficients:
1 | 1 -16 0 0 -40

Step3: Bring down first coefficient

Bring down 1:
1 | 1 -16 0 0 -40
1
1

Step4: Multiply and add (x³ term)

$1×1=1$; $-16+1=-15$:
1 | 1 -16 0 0 -40
1 -15
1 -15

Step5: Multiply and add (x² term)

$1×(-15)=-15$; $0+(-15)=-15$:
1 | 1 -16 0 0 -40
1 -15 -15
1 -15 -15

Step6: Multiply and add (x term)

$1×(-15)=-15$; $0+(-15)=-15$:
1 | 1 -16 0 0 -40
1 -15 -15 -15
1 -15 -15 -15

Step7: Multiply and add (remainder)

$1×(-15)=-15$; $-40+(-15)=-55$:
1 | 1 -16 0 0 -40
1 -15 -15 -15
1 -15 -15 -15 -55

Step8: Form the result

Quotient: $x^3 -15x^2 -15x -15$, remainder: $-55$.
Result: $x^3 -15x^2 -15x -15 + \frac{-55}{x-1}$