factor x^4-14x^3+71x^2-154x+120

1) Let's factor in

ℝ

x^{4}

- 14 x^{3}

+ 71 x^{2}

- 154 x

+ 120

⇥1.1) Try the obvious solution first:

x = 2

⇥Let's evaluate the value of:

2^{4}

- 14 \cdot 2^{3}

+ 71 \cdot 2^{2}

- 154 \cdot 2

+ 120

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

2^{4}

2 \cdot 2 \cdot 2 \cdot 2

16

2^{3}

2 \cdot 2 \cdot 2

8

2^{2}

2 \cdot 2

4

■Multiplications are done before additions and subtractions:

154 \cdot 2

(100 + 50 + 4) \cdot 2

100 \cdot 2 + 50 \cdot 2 + 4 \cdot 2

200 + 100 + 8

308

⇥

16

- 14 \cdot 8

+ 71 \cdot 4

- 308

+ 120

⇥ⓘ Multiplications are done before additions and subtractions:

14 \cdot 8

(10 + 4) \cdot 8

10 \cdot 8 + 4 \cdot 8

80 + 32

112

71 \cdot 4

(70 + 1) \cdot 4

70 \cdot 4 + 4

280 + 4

284

⇥

16

- 112

+ 284

- 308

+ 120

⇥ⓘ Start from

16

and move back

112

on the number line, going below zero:

16 - 112

-96

⇥

- 96

+ 284

- 308

+ 120

⇥ⓘ Start from

-96

and move forward

284

on the number line, going past zero:

- 96 + 284

188

⇥

188 - 308 + 120

⇥ⓘ Start from

188

and move back

308

on the number line, going below zero:

188 - 308

-120

⇥

- 120 + 120

⇥ⓘ Start from

-120

and move forward

120

on the number line, up to zero:

- 120 + 120

0

⇥

0

⇥Root found:

x = 2

⇥1.2) Compute the quotient and remainder of

(x^{4}

- 14 x^{3}

+ 71 x^{2}

- 154 x

+ 120)

(x - 2)

⇥

Expression in progress	Divide leading terms	Quotient	Quotient × divisor	Remainder after subtracting
$x^{4}$ $- 14 x^{3}$ $+ 71 x^{2}$ $- 154 x$ $+ 120$	$\frac{x^{4}}{x}$	$x^{3}$	$x^{3} \cdot (x - 2) = x^{4} - 2 x^{3}$	$(x^{4}$ $- 14 x^{3}$ $+ 71 x^{2}$ $- 154 x$ $+ 120) - (x^{4} - 2 x^{3}) = - 12 x^{3}$ $+ 71 x^{2}$ $- 154 x$ $+ 120$
$- 12 x^{3}$ $+ 71 x^{2}$ $- 154 x$ $+ 120$	$- \frac{12 x^{3}}{x}$	$- 12 x^{2}$	$- 12 x^{2} \cdot (x - 2) = - 12 x^{3} + 24 x^{2}$	$(- 12 x^{3}$ $+ 71 x^{2}$ $- 154 x$ $+ 120) - (- 12 x^{3} + 24 x^{2}) = 47 x^{2} - 154 x + 120$
$47 x^{2} - 154 x + 120$	$\frac{47 x^{2}}{x}$	$47 x$	$47 x \cdot (x - 2) = 47 x^{2} - 94 x$	$(47 x^{2} - 154 x + 120) - (47 x^{2} - 94 x) = - 60 x + 120$
$- 60 x + 120$	$- \frac{60 x}{x}$	$-60$	$-60 \cdot (x - 2) = - 60 x + 120$	$(- 60 x + 120) - (- 60 x + 120) = 0$

⇥So the result of the euclidean division is:

x^{3}

- 12 x^{2}

+ 47 x

- 60

⇥1.3) Let's factorize in

ℝ

(x^{3}

- 12 x^{2}

+ 47 x

- 60)

⇥⇥1.3.1) Try the obvious solution first:

x = 3

⇥⇥Let's evaluate the value of:

3^{3}

- 12 \cdot 3^{2}

+ 47 \cdot 3

- 60

⇥⇥ⓘ 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:

47 \cdot 3

(40 + 7) \cdot 3

40 \cdot 3 + 7 \cdot 3

120 + 21

141

⇥⇥

27

- 12 \cdot 9

+ 141

- 60

⇥⇥ⓘ Multiplications are done before additions and subtractions:

12 \cdot 9

(10 + 2) \cdot 9

10 \cdot 9 + 2 \cdot 9

90 + 18

108

⇥⇥

27

- 108

+ 141

- 60

⇥⇥ⓘ Start from

27

and move back

108

on the number line, going below zero:

27 - 108

-81

⇥⇥

- 81 + 141 - 60

⇥⇥ⓘ Start from

-81

and move forward

141

on the number line, going past zero:

- 81 + 141

60

⇥⇥

60 - 60

⇥⇥ⓘ Start from

60

and move back

60

on the number line, down to zero:

60 - 60

0

⇥⇥

0

⇥⇥Root found:

x = 3

⇥⇥1.3.2) Perform polynomial long division of

(x^{3}

- 12 x^{2}

+ 47 x

- 60)

(x - 3)

⇥⇥

Expression in progress	Divide leading terms	Quotient	Quotient × divisor	Remainder after subtracting
$x^{3}$ $- 12 x^{2}$ $+ 47 x$ $- 60$	$\frac{x^{3}}{x}$	$x^{2}$	$x^{2} \cdot (x - 3) = x^{3} - 3 x^{2}$	$(x^{3}$ $- 12 x^{2}$ $+ 47 x$ $- 60) - (x^{3} - 3 x^{2}) = - 9 x^{2} + 47 x - 60$
$- 9 x^{2} + 47 x - 60$	$- \frac{9 x^{2}}{x}$	$- 9 x$	$- 9 x \cdot (x - 3) = - 9 x^{2} + 27 x$	$(- 9 x^{2} + 47 x - 60) - (- 9 x^{2} + 27 x) = 20 x - 60$
$20 x - 60$	$\frac{20 x}{x}$	$20$	$20 \cdot (x - 3) = 20 x - 60$	$(20 x - 60) - (20 x - 60) = 0$

⇥⇥So the result of the euclidean division is:

x^{2} - 9 x + 20

⇥⇥1.3.3) Let's factorize in

ℝ

(x^{2} - 9 x + 20)

⇥⇥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 = -9

and

c = 20

⇥⇥

Δ = {(-9)}^{2} - 4 \cdot 1 \cdot 20

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

{(-9)}^{2}

(-9) \cdot (-9)

81

⇥⇥

Δ = 81 - 4 \cdot 20

⇥⇥ⓘ Multiplications are done before additions and subtractions:

4 \cdot 20

80

⇥⇥

Δ = 81 - 80

⇥⇥ⓘ Start from

81

and move back

80

on the number line:

81 - 80

1

⇥⇥

Δ = 1

⇥⇥As

Δ > 0

, the equation has two roots:

⇥⇥The quadratic formula gives

x_{1}

\frac{- b - \sqrt{Δ}}{2 a}

\frac{- (-9) - \sqrt{1}}{2 \cdot 1}

⇥⇥

x_{1} = \frac{9 - 1}{2}

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

9 - 1

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{- (-9) + \sqrt{1}}{2 \cdot 1}

⇥⇥

x_{2} = \frac{9 + 1}{2}

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

9 + 1

10

⇥⇥

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

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

2

, their greatest common divisor:

\frac{10}{2}

5

⇥⇥

x_{2} = 5

⇥⇥Factors identified:

(x - 4) \cdot (x - 5)

▷The factor

(x^{3}

- 12 x^{2}

+ 47 x

- 60)

factors as:

(x - 3) \cdot (x - 4) \cdot (x - 5)

▶Factored expression:

(x - 2) \cdot (x - 3) \cdot (x - 4) \cdot (x - 5)