factor x^3+6x^2-4x-24 — step-by-step solution

1) Let's factor in

ℝ

x^{3}

+ 6 x^{2}

- 4 x

- 24

⇥1.1) Try the obvious solution first:

x = 2

⇥Let's evaluate the value of:

2^{3}

+ 6 \cdot 2^{2}

- 4 \cdot 2

- 24

⇥ⓘ 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:

4 \cdot 2

8

⇥

8

+ 6 \cdot 4

- 8

- 24

⇥ⓘ Multiplications are done before additions and subtractions:

6 \cdot 4

24

⇥

8

+ 24

- 8

- 24

⇥ⓘ Start from

8

and move forward

24

on the number line:

8 + 24

32

⇥

32 - 8 - 24

⇥ⓘ Start from

32

and move back

8

on the number line:

32 - 8

24

⇥

24 - 24

⇥ⓘ Start from

24

and move back

24

on the number line, down to zero:

24 - 24

0

⇥

0

⇥Root found:

x = 2

⇥1.2) Perform polynomial long division of

(x^{3}

+ 6 x^{2}

- 4 x

- 24)

(x - 2)

⇥

Expression in progress	Divide leading terms	Quotient	Quotient × divisor	Remainder after subtracting
$x^{3}$ $+ 6 x^{2}$ $- 4 x$ $- 24$	$\frac{x^{3}}{x}$	$x^{2}$	$x^{2} \cdot (x - 2) = x^{3} - 2 x^{2}$	$(x^{3}$ $+ 6 x^{2}$ $- 4 x$ $- 24) - (x^{3} - 2 x^{2}) = 8 x^{2} - 4 x - 24$
$8 x^{2} - 4 x - 24$	$\frac{8 x^{2}}{x}$	$8 x$	$8 x \cdot (x - 2) = 8 x^{2} - 16 x$	$(8 x^{2} - 4 x - 24) - (8 x^{2} - 16 x) = 12 x - 24$
$12 x - 24$	$\frac{12 x}{x}$	$12$	$12 \cdot (x - 2) = 12 x - 24$	$(12 x - 24) - (12 x - 24) = 0$

⇥Therefore, the result of the euclidean division is:

x^{2} + 8 x + 12

⇥1.3) Let's factorize in

ℝ

(x^{2} + 8 x + 12)

⇥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 = 8

and

c = 12

⇥

Δ = 8^{2} - 4 \cdot 1 \cdot 12

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

8^{2}

8 \cdot 8

64

⇥

Δ = 64 - 4 \cdot 12

⇥ⓘ Multiplications are done before additions and subtractions:

4 \cdot 12

4 \cdot (10 + 2)

4 \cdot 10 + 4 \cdot 2

40 + 8

48

⇥

Δ = 64 - 48

⇥ⓘ Start from

64

and move back

48

on the number line:

64 - 48

16

⇥

Δ = 16

⇥Since

Δ > 0

, the equation has two roots:

⇥The quadratic formula gives

x_{1}

\frac{- b - \sqrt{Δ}}{2 a}

\frac{- 8 - \sqrt{16}}{2 \cdot 1}

⇥

x_{1} = \frac{- 8 - \sqrt{16}}{2}

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

\sqrt{16}

4

;

4^{2}

16

⇥

x_{1} = \frac{- 8 - 4}{2}

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

- 8 - 4

-12

⇥

x_{1} = - \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_{1} = -6

⇥The quadratic formula gives

x_{2}

\frac{- b + \sqrt{Δ}}{2 a}

\frac{- 8 + \sqrt{16}}{2 \cdot 1}

⇥

x_{2} = \frac{- 8 + \sqrt{16}}{2}

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

\sqrt{16}

4

;

4^{2}

16

⇥

x_{2} = \frac{- 8 + 4}{2}

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

- 8 + 4

-4

⇥

x_{2} = - \frac{4}{2}

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

2

, their greatest common divisor:

- \frac{4}{2}

-2

⇥

x_{2} = -2

▷The factor

(x^{2} + 8 x + 12)

factors as:

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

▶Factored expression:

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