factorize x^4-4x^3-7x^2+22x+24 — step-by-step solution

1) Let's factor in

ℝ

x^{4}

- 4 x^{3}

- 7 x^{2}

+ 22 x

+ 24

⇥1.1) Let's test the obvious candidate:

x = -1

⇥Let's evaluate the value of:

{(-1)}^{4}

- 4 \cdot {(-1)}^{3}

- 7 \cdot {(-1)}^{2}

+ 22 \cdot (-1)

+ 24

⇥ⓘ Calculate powers before multiplications, additions and subtractions. With a positive integer exponent, a power corresponds to a repeated product:

{(-1)}^{4}

(-1) \cdot (-1) \cdot (-1) \cdot (-1)

1

■Powers are computed before multiplications, additions and subtractions. With a positive integer exponent, a power corresponds to a repeated product:

{(-1)}^{3}

(-1) \cdot (-1) \cdot (-1)

-1

■Multiplication by -1 gives the opposite value:

- 4 \cdot (-1)

4

■Calculate powers before multiplications, additions and subtractions. A power with a positive integer exponent is a repeated product:

{(-1)}^{2}

(-1) \cdot (-1)

1

■Multiplying by 1 changes nothing:

7 \cdot 1

7

■Multiplications are done before additions and subtractions. A single minus sign does not cancel out, the product of a negative number by a positive number is negative:

22 \cdot (-1)

-22

⇥

1

+ 4

- 7

- 22

+ 24

⇥ⓘ Start from

1

and move forward

4

on the number line:

1 + 4

5

⇥

5

- 7

- 22

+ 24

⇥ⓘ Start from

5

and move back

7

on the number line, going below zero:

5 - 7

-2

⇥

- 2 - 22 + 24

⇥ⓘ Start from

-2

and move back

22

more on the number line, away from zero:

- 2 - 22

-24

⇥

- 24 + 24

⇥ⓘ Start from

-24

and move forward

24

on the number line, up to zero:

- 24 + 24

0

⇥

0

⇥We find the root:

x = -1

⇥1.2) Let's start the Euclidean division of

(x^{4}

- 4 x^{3}

- 7 x^{2}

+ 22 x

+ 24)

(x + 1)

⇥

Current expression	Divide leading terms	Quotient	Quotient × divisor	Remainder after subtracting
$x^{4}$ $- 4 x^{3}$ $- 7 x^{2}$ $+ 22 x$ $+ 24$	$\frac{x^{4}}{x}$	$x^{3}$	$x^{3} \cdot (x + 1) = x^{4} + x^{3}$	$(x^{4}$ $- 4 x^{3}$ $- 7 x^{2}$ $+ 22 x$ $+ 24) - (x^{4} + x^{3}) = - 5 x^{3}$ $- 7 x^{2}$ $+ 22 x$ $+ 24$
$- 5 x^{3}$ $- 7 x^{2}$ $+ 22 x$ $+ 24$	$- \frac{5 x^{3}}{x}$	$- 5 x^{2}$	$- 5 x^{2} \cdot (x + 1) = - 5 x^{3} - 5 x^{2}$	$(- 5 x^{3}$ $- 7 x^{2}$ $+ 22 x$ $+ 24) - (- 5 x^{3} - 5 x^{2}) = - 2 x^{2} + 22 x + 24$
$- 2 x^{2} + 22 x + 24$	$- \frac{2 x^{2}}{x}$	$- 2 x$	$- 2 x \cdot (x + 1) = - 2 x^{2} - 2 x$	$(- 2 x^{2} + 22 x + 24) - (- 2 x^{2} - 2 x) = 24 x + 24$
$24 x + 24$	$\frac{24 x}{x}$	$24$	$24 \cdot (x + 1) = 24 x + 24$	$(24 x + 24) - (24 x + 24) = 0$

⇥So the result of the Euclidean division is:

x^{3}

- 5 x^{2}

- 2 x

+ 24

⇥1.3) Let's factorize in

ℝ

(x^{3}

- 5 x^{2}

- 2 x

+ 24)

⇥⇥1.3.1) Let's test the obvious candidate:

x = -2

⇥⇥Let's compute the value of:

{(-2)}^{3}

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

- 2 \cdot (-2)

+ 24

⇥⇥ⓘ Calculate powers before multiplications, additions and subtractions. A power with a positive integer exponent is a repeated product:

{(-2)}^{3}

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

-8

■Calculate powers before multiplications, additions and subtractions. With a positive integer exponent, a power corresponds to a repeated product:

{(-2)}^{2}

(-2) \cdot (-2)

4

■Multiplications are done before additions and subtractions. In a product, two minus signs cancel: multiplying two negative numbers gives a positive number:

- 2 \cdot (-2)

4

⇥⇥

- 8

- 5 \cdot 4

+ 4

+ 24

⇥⇥ⓘ Perform multiplications before additions and subtractions:

5 \cdot 4

20

⇥⇥

- 8

- 20

+ 4

+ 24

⇥⇥ⓘ Start from

-8

and move back

20

more on the number line, away from zero:

- 8 - 20

-28

⇥⇥

- 28 + 4 + 24

⇥⇥ⓘ Start from

-28

and move forward

4

on the number line, getting closer to zero:

- 28 + 4

-24

⇥⇥

- 24 + 24

⇥⇥ⓘ Start from

-24

and move forward

24

on the number line, up to zero:

- 24 + 24

0

⇥⇥

0

⇥⇥We find the root:

x = -2

⇥⇥1.3.2) Compute the quotient and remainder of

(x^{3}

- 5 x^{2}

- 2 x

+ 24)

(x + 2)

⇥⇥

Expression in progress	Divide leading terms	Quotient	Quotient × divisor	Remainder after subtracting
$x^{3}$ $- 5 x^{2}$ $- 2 x$ $+ 24$	$\frac{x^{3}}{x}$	$x^{2}$	$x^{2} \cdot (x + 2) = x^{3} + 2 x^{2}$	$(x^{3}$ $- 5 x^{2}$ $- 2 x$ $+ 24) - (x^{3} + 2 x^{2}) = - 7 x^{2} - 2 x + 24$
$- 7 x^{2} - 2 x + 24$	$- \frac{7 x^{2}}{x}$	$- 7 x$	$- 7 x \cdot (x + 2) = - 7 x^{2} - 14 x$	$(- 7 x^{2} - 2 x + 24) - (- 7 x^{2} - 14 x) = 12 x + 24$
$12 x + 24$	$\frac{12 x}{x}$	$12$	$12 \cdot (x + 2) = 12 x + 24$	$(12 x + 24) - (12 x + 24) = 0$

⇥⇥So the result of the Euclidean division is:

x^{2} - 7 x + 12

⇥⇥1.3.3) Let's factorize in

ℝ

(x^{2} - 7 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 = -7

and

c = 12

⇥⇥

Δ = {(-7)}^{2} - 4 \cdot 1 \cdot 12

⇥⇥ⓘ Calculate powers before multiplications, additions and subtractions. With a positive integer exponent, a power corresponds to a repeated product:

{(-7)}^{2}

(-7) \cdot (-7)

49

⇥⇥

Δ = 49 - 4 \cdot 12

⇥⇥ⓘ Perform multiplications before additions and subtractions:

4 \cdot 12

4 \cdot (10 + 2)

4 \cdot 10 + 4 \cdot 2

40 + 8

48

⇥⇥

Δ = 49 - 48

⇥⇥ⓘ Start from

49

and move back

48

on the number line:

49 - 48

1

⇥⇥

Δ = 1

⇥⇥As

Δ > 0

, the equation has two roots:

⇥⇥The quadratic formula gives

x_{1}

\frac{- b - \sqrt{Δ}}{2 a}

\frac{- (-7) - \sqrt{1}}{2 \cdot 1}

⇥⇥

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

⇥⇥ⓘ Calculate the numerator and denominator before dividing:

7 - 1

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{- (-7) + \sqrt{1}}{2 \cdot 1}

⇥⇥

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

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

7 + 1

8

⇥⇥

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

⇥⇥ⓘ Simplify the fraction by dividing the numerator and the denominator by

2

, their greatest common divisor:

\frac{8}{2}

4

⇥⇥

x_{2} = 4

⇥⇥Factors found:

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

▷The factor

(x^{3}

- 5 x^{2}

- 2 x

+ 24)

factors as:

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

▶Factored expression:

(x + 1) \cdot (x + 2) \cdot (x - 3) \cdot (x - 4)