factor x^3+2x^2-36x-72 — step-by-step solution

1) Let's factorize in

ℝ

x^{3}

+ 2 x^{2}

- 36 x

- 72

⇥1.1) Try the obvious solution first:

x = -2

⇥Let's evaluate the value of:

{(-2)}^{3}

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

- 36 \cdot (-2)

- 72

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

{(-2)}^{3}

(-2) \cdot (-2) \cdot (-2)

-8

{(-2)}^{2}

(-2) \cdot (-2)

4

■Multiplications are done before additions and subtractions. Two minus signs cancel out, the product of two negative numbers is positive:

- 36 \cdot (-2)

36 \cdot 2

(30 + 6) \cdot 2

30 \cdot 2 + 6 \cdot 2

60 + 12

72

⇥

- 8

+ 2 \cdot 4

+ 72

- 72

⇥ⓘ Multiplications are done before additions and subtractions:

2 \cdot 4

8

■Start from

72

and move back

72

on the number line, down to zero:

72 - 72

0

⇥

- 8 + 8 + 0

⇥ⓘ Adding 0 changes nothing:

8 + 0

8

■Start from

-8

and move forward

8

on the number line, up to zero:

- 8 + 8

0

⇥

0

⇥Root found:

x = -2

⇥1.2) Compute the quotient and remainder of

(x^{3}

+ 2 x^{2}

- 36 x

- 72)

(x + 2)

⇥

Expression in progress	Divide leading terms	Quotient	Quotient × divisor	Remainder after subtracting
$x^{3}$ $+ 2 x^{2}$ $- 36 x$ $- 72$	$\frac{x^{3}}{x}$	$x^{2}$	$x^{2} \cdot (x + 2) = x^{3} + 2 x^{2}$	$(x^{3}$ $+ 2 x^{2}$ $- 36 x$ $- 72) - (x^{3} + 2 x^{2}) = - 36 x - 72$
$- 36 x - 72$	$- \frac{36 x}{x}$	$-36$	$-36 \cdot (x + 2) = - 36 x - 72$	$(- 36 x - 72) - (- 36 x - 72) = 0$

⇥So the result of the euclidean division is:

x^{2} - 36

⇥1.3) Let's factorize in

ℝ

(x^{2} - 36)

⇥For

x^{2} - 36

, we recognize a difference of two squares:

x^{2} - 6^{2}

. Apply

a^{2} - b^{2} = (a - b) \cdot (a + b)

, with

a = x

and

b = 6

▷The factor

(x^{2} - 36)

factors as:

(x + 6) \cdot (x - 6)

▶Factored expression:

(x + 2) \cdot (x + 6) \cdot (x - 6)