factor x^3-3x^2-8x-10 — step-by-step solution

1) Let's factor in

ℝ

x^{3}

- 3 x^{2}

- 8 x

- 10

⇥1.1) Let's start with the obvious candidate:

x = 5

⇥Let's compute the value of:

5^{3}

- 3 \cdot 5^{2}

- 8 \cdot 5

- 10

⇥ⓘ Powers are computed before multiplications, additions and subtractions. An integer power is a repeated product:

5^{3}

5 \cdot 5 \cdot 5

125

5^{2}

5 \cdot 5

25

■Multiplications are done before additions and subtractions:

8 \cdot 5

40

⇥

125

- 3 \cdot 25

- 40

- 10

⇥ⓘ Multiplications are done before additions and subtractions:

3 \cdot 25

3 \cdot (20 + 5)

3 \cdot 20 + 3 \cdot 5

60 + 15

75

⇥

125

- 75

- 40

- 10

⇥ⓘ Start from

125

and move back

75

on the number line:

125 - 75

50

⇥

50 - 40 - 10

⇥ⓘ Start from

50

and move back

40

on the number line:

50 - 40

10

⇥

10 - 10

⇥ⓘ Start from

10

and move back

10

on the number line, down to zero:

10 - 10

0

⇥

0

⇥Root found:

x = 5

⇥1.2) Let's start the euclidean division of

(x^{3}

- 3 x^{2}

- 8 x

- 10)

(x - 5)

⇥

Current expression	Divide leading terms	Quotient	Quotient × divisor	Remainder after subtracting
$x^{3}$ $- 3 x^{2}$ $- 8 x$ $- 10$	$\frac{x^{3}}{x}$	$x^{2}$	$x^{2} \cdot (x - 5) = x^{3} - 5 x^{2}$	$(x^{3}$ $- 3 x^{2}$ $- 8 x$ $- 10) - (x^{3} - 5 x^{2}) = 2 x^{2} - 8 x - 10$
$2 x^{2} - 8 x - 10$	$\frac{2 x^{2}}{x}$	$2 x$	$2 x \cdot (x - 5) = 2 x^{2} - 10 x$	$(2 x^{2} - 8 x - 10) - (2 x^{2} - 10 x) = 2 x - 10$
$2 x - 10$	$\frac{2 x}{x}$	$2$	$2 \cdot (x - 5) = 2 x - 10$	$(2 x - 10) - (2 x - 10) = 0$

⇥Therefore, the result of the euclidean division is:

x^{2} + 2 x + 2

ⅰ

x^{3}

- 3 x^{2}

- 8 x

- 10

(x - 5) \cdot (x^{2} + 2 x + 2)

▶Factored expression:

(x - 5) \cdot (x^{2} + 2 x + 2)