Question

divide.
use long division

1. ((x^3 - 10x^2 + 7x) div (x - 1))

use synthetic

1. ((n^3 - 4n^2 + 9) div (n - 4))

find all roots.

1. (x^3 + 5x^2 - 5x - 25 = 0)
2. (x^3 - 5x^2 - 4x + 20 = 0)
3. (x^3 - x^2 + 2x - 2 = 0)
4. (x^4 - 4x^2 + 3 = 0)

find all the zeros
aproximate the relative maxima and relative minimun
state the intervals of increase and decrease

1. (f(x) = x^3 - x^2)
2. (f(x) = x^4 - 4x^2 + 4)

Explanation:

Problem 19: $(x^3 - 10x^2 + 7x) \div (x - 1)$ (Long Division)

Step1: Divide leading terms

$\frac{x^3}{x} = x^2$
Multiply $(x-1)$ by $x^2$: $x^3 - x^2$
Subtract from dividend:
$(x^3 -10x^2 +7x) - (x^3 -x^2) = -9x^2 +7x$

Step2: Divide new leading terms

$\frac{-9x^2}{x} = -9x$
Multiply $(x-1)$ by $-9x$: $-9x^2 +9x$
Subtract:
$(-9x^2 +7x) - (-9x^2 +9x) = -2x$

Step3: Divide new leading terms

$\frac{-2x}{x} = -2$
Multiply $(x-1)$ by $-2$: $-2x +2$
Subtract:
$(-2x) - (-2x +2) = -2$

Step1: Set up synthetic division

Root of divisor: $n=4$
Coefficients of dividend: $1, -4, 0, 9$ (0 for missing $n$ term)

Step2: Carry down and multiply

Carry down 1. Multiply by 4: $1 \times 4 = 4$
Add to next coefficient: $-4 + 4 = 0$

Step3: Multiply and add

Multiply 0 by 4: $0 \times 4 = 0$
Add to next coefficient: $0 + 0 = 0$

Step4: Multiply and add

Multiply 0 by 4: $0 \times 4 = 0$
Add to last coefficient: $9 + 0 = 9$

Step1: Factor by grouping

Group terms: $(x^3 +5x^2) + (-5x -25) = 0$
Factor groups: $x^2(x+5) -5(x+5) = 0$
Factor out $(x+5)$: $(x+5)(x^2 -5) = 0$

Step2: Solve for $x$

Set each factor to 0:
$x+5=0 \implies x=-5$
$x^2-5=0 \implies x=\pm\sqrt{5}$

Answer:

$x^2 -9x -2 - \frac{2}{x-1}$

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Problem 20: $(n^3 - 4n^2 + 9) \div (n - 4)$ (Synthetic Division)