factor x^4-6x^3+9x^2+4x-12 — step-by-step solution

1) Let's factor in

ℝ

x^{4}

- 6 x^{3}

+ 9 x^{2}

+ 4 x

- 12

⇥1.1) Let's start with the obvious candidate:

x = -1

⇥Let's evaluate the value of:

{(-1)}^{4}

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

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

+ 4 \cdot (-1)

- 12

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

{(-1)}^{4}

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

1

{(-1)}^{3}

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

-1

{(-1)}^{2}

(-1) \cdot (-1)

1

■Multiplying by -1 gives the opposite:

- 6 \cdot (-1)

6

■Multiplying by 1 changes nothing:

9 \cdot 1

9

■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:

4 \cdot (-1)

-4

⇥

1

+ 6

+ 9

- 4

- 12

⇥ⓘ Start from

1

and move forward

6

on the number line:

1 + 6

7

⇥

7

+ 9

- 4

- 12

⇥ⓘ Start from

7

and move forward

9

on the number line:

7 + 9

16

⇥

16 - 4 - 12

⇥ⓘ Start from

16

and move back

4

on the number line:

16 - 4

12

⇥

12 - 12

⇥ⓘ Start from

12

and move back

12

on the number line, down to zero:

12 - 12

0

⇥

0

⇥Root found:

x = -1

⇥1.2) Let's start the euclidean division of

(x^{4}

- 6 x^{3}

+ 9 x^{2}

+ 4 x

- 12)

(x + 1)

⇥

Expression in progress	Divide leading terms	Quotient	Quotient × divisor	Remainder after subtracting
$x^{4}$ $- 6 x^{3}$ $+ 9 x^{2}$ $+ 4 x$ $- 12$	$\frac{x^{4}}{x}$	$x^{3}$	$x^{3} \cdot (x + 1) = x^{4} + x^{3}$	$(x^{4}$ $- 6 x^{3}$ $+ 9 x^{2}$ $+ 4 x$ $- 12) - (x^{4} + x^{3}) = - 7 x^{3}$ $+ 9 x^{2}$ $+ 4 x$ $- 12$
$- 7 x^{3}$ $+ 9 x^{2}$ $+ 4 x$ $- 12$	$- \frac{7 x^{3}}{x}$	$- 7 x^{2}$	$- 7 x^{2} \cdot (x + 1) = - 7 x^{3} - 7 x^{2}$	$(- 7 x^{3}$ $+ 9 x^{2}$ $+ 4 x$ $- 12) - (- 7 x^{3} - 7 x^{2}) = 16 x^{2} + 4 x - 12$
$16 x^{2} + 4 x - 12$	$\frac{16 x^{2}}{x}$	$16 x$	$16 x \cdot (x + 1) = 16 x^{2} + 16 x$	$(16 x^{2} + 4 x - 12) - (16 x^{2} + 16 x) = - 12 x - 12$
$- 12 x - 12$	$- \frac{12 x}{x}$	$-12$	$-12 \cdot (x + 1) = - 12 x - 12$	$(- 12 x - 12) - (- 12 x - 12) = 0$

⇥So the result of the euclidean division is:

x^{3}

- 7 x^{2}

+ 16 x

- 12

⇥1.3) Let's factor in

ℝ

(x^{3}

- 7 x^{2}

+ 16 x

- 12)

⇥⇥1.3.1) Let's start with the obvious candidate:

x = 2

⇥⇥Let's evaluate the value of:

2^{3}

- 7 \cdot 2^{2}

+ 16 \cdot 2

- 12

⇥⇥ⓘ 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:

16 \cdot 2

(10 + 6) \cdot 2

10 \cdot 2 + 6 \cdot 2

20 + 12

32

⇥⇥

8

- 7 \cdot 4

+ 32

- 12

⇥⇥ⓘ Multiplications are done before additions and subtractions:

7 \cdot 4

28

⇥⇥

8

- 28

+ 32

- 12

⇥⇥ⓘ Start from

8

and move back

28

on the number line, going below zero:

8 - 28

-20

⇥⇥

- 20 + 32 - 12

⇥⇥ⓘ Start from

-20

and move forward

32

on the number line, going past zero:

- 20 + 32

12

⇥⇥

12 - 12

⇥⇥ⓘ Start from

12

and move back

12

on the number line, down to zero:

12 - 12

0

⇥⇥

0

⇥⇥Root found:

x = 2

⇥⇥1.3.2) Let's start the euclidean division of

(x^{3}

- 7 x^{2}

+ 16 x

- 12)

(x - 2)

⇥⇥

Current expression	Divide leading terms	Quotient	Quotient × divisor	Remainder after subtracting
$x^{3}$ $- 7 x^{2}$ $+ 16 x$ $- 12$	$\frac{x^{3}}{x}$	$x^{2}$	$x^{2} \cdot (x - 2) = x^{3} - 2 x^{2}$	$(x^{3}$ $- 7 x^{2}$ $+ 16 x$ $- 12) - (x^{3} - 2 x^{2}) = - 5 x^{2} + 16 x - 12$
$- 5 x^{2} + 16 x - 12$	$- \frac{5 x^{2}}{x}$	$- 5 x$	$- 5 x \cdot (x - 2) = - 5 x^{2} + 10 x$	$(- 5 x^{2} + 16 x - 12) - (- 5 x^{2} + 10 x) = 6 x - 12$
$6 x - 12$	$\frac{6 x}{x}$	$6$	$6 \cdot (x - 2) = 6 x - 12$	$(6 x - 12) - (6 x - 12) = 0$

⇥⇥So the result of the euclidean division is:

x^{2} - 5 x + 6

⇥⇥1.3.3) Let's factorize in

ℝ

(x^{2} - 5 x + 6)

⇥⇥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 = -5

and

c = 6

⇥⇥

Δ = {(-5)}^{2} - 4 \cdot 1 \cdot 6

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

{(-5)}^{2}

(-5) \cdot (-5)

25

⇥⇥

Δ = 25 - 4 \cdot 6

⇥⇥ⓘ Multiplications are done before additions and subtractions:

4 \cdot 6

24

⇥⇥

Δ = 25 - 24

⇥⇥ⓘ Start from

25

and move back

24

on the number line:

25 - 24

1

⇥⇥

Δ = 1

⇥⇥Since

Δ > 0

, the equation has two roots:

⇥⇥The quadratic formula gives

x_{1}

\frac{- b - \sqrt{Δ}}{2 a}

\frac{- (-5) - \sqrt{1}}{2 \cdot 1}

⇥⇥

x_{1} = \frac{5 - 1}{2}

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

5 - 1

4

⇥⇥

x_{1} = \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_{1} = 2

⇥⇥The quadratic formula gives

x_{2}

\frac{- b + \sqrt{Δ}}{2 a}

\frac{- (-5) + \sqrt{1}}{2 \cdot 1}

⇥⇥

x_{2} = \frac{5 + 1}{2}

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

5 + 1

6

⇥⇥

x_{2} = \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_{2} = 3

⇥⇥Factors identified:

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

▷The factor

(x^{3}

- 7 x^{2}

+ 16 x

- 12)

factors as:

{(x - 2)}^{2} \cdot (x - 3)

▶Factored expression:

(x + 1) \cdot (x - 3) \cdot {(x - 2)}^{2}