factor x^3-3x^2-10x+24 — step-by-step solution

1) Let's factor in

ℝ

x^{3}

- 3 x^{2}

- 10 x

+ 24

⇥1.1) Try the obvious solution first:

x = 2

⇥Let's evaluate the value of:

2^{3}

- 3 \cdot 2^{2}

- 10 \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:

10 \cdot 2

20

⇥

8

- 3 \cdot 4

- 20

+ 24

⇥ⓘ Multiplications are done before additions and subtractions:

3 \cdot 4

12

⇥

8

- 12

- 20

+ 24

⇥ⓘ Start from

8

and move back

12

on the number line, going below zero:

8 - 12

-4

⇥

- 4 - 20 + 24

⇥ⓘ Start from

-4

and move back

20

more on the number line, away from zero:

- 4 - 20

-24

⇥

- 24 + 24

⇥ⓘ Start from

-24

and move forward

24

on the number line, up to zero:

- 24 + 24

0

⇥

0

⇥Root found:

x = 2

⇥1.2) Compute the quotient and remainder of

(x^{3}

- 3 x^{2}

- 10 x

+ 24)

(x - 2)

⇥

Expression in progress	Divide leading terms	Quotient	Quotient × divisor	Remainder after subtracting
$x^{3}$ $- 3 x^{2}$ $- 10 x$ $+ 24$	$\frac{x^{3}}{x}$	$x^{2}$	$x^{2} \cdot (x - 2) = x^{3} - 2 x^{2}$	$(x^{3}$ $- 3 x^{2}$ $- 10 x$ $+ 24) - (x^{3} - 2 x^{2}) = - x^{2} - 10 x + 24$
$- x^{2} - 10 x + 24$	$- \frac{x^{2}}{x}$	$- x$	$- x \cdot (x - 2) = - x^{2} + 2 x$	$(- x^{2} - 10 x + 24) - (- x^{2} + 2 x) = - 12 x + 24$
$- 12 x + 24$	$- \frac{12 x}{x}$	$-12$	$-12 \cdot (x - 2) = - 12 x + 24$	$(- 12 x + 24) - (- 12 x + 24) = 0$

⇥So the result of the euclidean division is:

x^{2} - x - 12

⇥1.3) Let's factorize in

ℝ

(x^{2} - x - 12)

⇥Given the quadratic form

a x^{2} + b x + c

, let's compute the discriminant:

Δ = b^{2} - 4 a c

. Here

a = 1

b = -1

and

c = -12

⇥

Δ = {(-1)}^{2} - 4 \cdot 1 \cdot (-12)

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

{(-1)}^{2}

(-1) \cdot (-1)

1

⇥

Δ = 1 - 4 \cdot (-12)

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

- 4 \cdot (-12)

4 \cdot 12

4 \cdot (10 + 2)

4 \cdot 10 + 4 \cdot 2

40 + 8

48

⇥

Δ = 1 + 48

⇥ⓘ Start from

1

and move forward

48

on the number line:

1 + 48

49

⇥

Δ = 49

⇥As

Δ > 0

, the equation has two roots:

⇥The quadratic formula gives

x_{1}

\frac{- b - \sqrt{Δ}}{2 a}

\frac{- (-1) - \sqrt{49}}{2 \cdot 1}

⇥

x_{1} = \frac{1 - \sqrt{49}}{2}

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

\sqrt{49}

7

;

7^{2}

49

⇥

x_{1} = \frac{1 - 7}{2}

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

1 - 7

-6

⇥

x_{1} = - \frac{6}{2}

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

2

, their greatest common divisor:

- \frac{6}{2}

-3

⇥

x_{1} = -3

⇥The quadratic formula gives

x_{2}

\frac{- b + \sqrt{Δ}}{2 a}

\frac{- (-1) + \sqrt{49}}{2 \cdot 1}

⇥

x_{2} = \frac{1 + \sqrt{49}}{2}

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

\sqrt{49}

7

;

7^{2}

49

⇥

x_{2} = \frac{1 + 7}{2}

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

1 + 7

8

⇥

x_{2} = \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_{2} = 4

▷The factor

(x^{2} - x - 12)

factors as:

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

▶Factored expression:

(x + 3) \cdot (x - 2) \cdot (x - 4)