factor x^3-5x^2-18x+72 — step-by-step solution

1) Let's factorize in

ℝ

x^{3}

- 5 x^{2}

- 18 x

+ 72

⇥1.1) Try the obvious solution first:

x = 3

⇥Let's compute the value of:

3^{3}

- 5 \cdot 3^{2}

- 18 \cdot 3

+ 72

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

3^{3}

3 \cdot 3 \cdot 3

27

3^{2}

3 \cdot 3

9

■Multiplications are done before additions and subtractions:

18 \cdot 3

(10 + 8) \cdot 3

10 \cdot 3 + 8 \cdot 3

30 + 24

54

⇥

27

- 5 \cdot 9

- 54

+ 72

⇥ⓘ Multiplications are done before additions and subtractions:

5 \cdot 9

45

⇥

27

- 45

- 54

+ 72

⇥ⓘ Start from

27

and move back

45

on the number line, going below zero:

27 - 45

-18

⇥

- 18 - 54 + 72

⇥ⓘ Start from

-18

and move back

54

more on the number line, away from zero:

- 18 - 54

-72

⇥

- 72 + 72

⇥ⓘ Start from

-72

and move forward

72

on the number line, up to zero:

- 72 + 72

0

⇥

0

⇥Root found:

x = 3

⇥1.2) Perform polynomial long division of

(x^{3}

- 5 x^{2}

- 18 x

+ 72)

(x - 3)

⇥

Current expression	Divide leading terms	Quotient	Quotient × divisor	Remainder after subtracting
$x^{3}$ $- 5 x^{2}$ $- 18 x$ $+ 72$	$\frac{x^{3}}{x}$	$x^{2}$	$x^{2} \cdot (x - 3) = x^{3} - 3 x^{2}$	$(x^{3}$ $- 5 x^{2}$ $- 18 x$ $+ 72) - (x^{3} - 3 x^{2}) = - 2 x^{2} - 18 x + 72$
$- 2 x^{2} - 18 x + 72$	$- \frac{2 x^{2}}{x}$	$- 2 x$	$- 2 x \cdot (x - 3) = - 2 x^{2} + 6 x$	$(- 2 x^{2} - 18 x + 72) - (- 2 x^{2} + 6 x) = - 24 x + 72$
$- 24 x + 72$	$- \frac{24 x}{x}$	$-24$	$-24 \cdot (x - 3) = - 24 x + 72$	$(- 24 x + 72) - (- 24 x + 72) = 0$

⇥So the result of the euclidean division is:

x^{2} - 2 x - 24

⇥1.3) Let's factor in

ℝ

(x^{2} - 2 x - 24)

⇥Given the quadratic form

a x^{2} + b x + c

, you can calculate the discriminant:

Δ = b^{2} - 4 a c

. Here

a = 1

b = -2

and

c = -24

⇥

Δ = {(-2)}^{2} - 4 \cdot 1 \cdot (-24)

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

{(-2)}^{2}

(-2) \cdot (-2)

4

⇥

Δ = 4 - 4 \cdot (-24)

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

- 4 \cdot (-24)

4 \cdot 24

4 \cdot (20 + 4)

4 \cdot 20 + 4 \cdot 4

80 + 16

96

⇥

Δ = 4 + 96

⇥ⓘ Start from

4

and move forward

96

on the number line:

4 + 96

100

⇥

Δ = 100

⇥Since

Δ > 0

, the equation has two roots:

⇥The quadratic formula gives

x_{1}

\frac{- b - \sqrt{Δ}}{2 a}

\frac{- (-2) - \sqrt{100}}{2 \cdot 1}

⇥

x_{1} = \frac{2 - \sqrt{100}}{2}

⇥ⓘ The square root of a number is the positive number whose square is that number:

\sqrt{100}

10

;

10^{2}

100

⇥

x_{1} = \frac{2 - 10}{2}

⇥ⓘ The numerator and the denominator are computed before the division:

2 - 10

-8

⇥

x_{1} = - \frac{8}{2}

⇥ⓘ We simplify the fraction by dividing the numerator and the denominator by

2

, their greatest common divisor:

- \frac{8}{2}

-4

⇥

x_{1} = -4

⇥The quadratic formula gives

x_{2}

\frac{- b + \sqrt{Δ}}{2 a}

\frac{- (-2) + \sqrt{100}}{2 \cdot 1}

⇥

x_{2} = \frac{2 + \sqrt{100}}{2}

⇥ⓘ The square root of a number is the positive number whose square is that number:

\sqrt{100}

10

;

10^{2}

100

⇥

x_{2} = \frac{2 + 10}{2}

⇥ⓘ The numerator and the denominator are computed before the division:

2 + 10

12

⇥

x_{2} = \frac{12}{2}

⇥ⓘ We simplify the fraction by dividing the numerator and the denominator by

2

, their greatest common divisor:

\frac{12}{2}

6

⇥

x_{2} = 6

▷The factor

(x^{2} - 2 x - 24)

factors as:

(x + 4) \cdot (x - 6)

▶Factored expression:

(x + 4) \cdot (x - 3) \cdot (x - 6)