factor -λ^3+3λ+2 — step-by-step solution

1) Let's factorize in

ℝ

- λ^{3} + 3 λ + 2

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

λ = -1

⇥Let's compute the value of:

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

⇥ⓘ An integer power is a repeated product:

- {(-1)}^{3}

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

- (-1)

■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:

3 \cdot (-1)

-3

⇥

- (-1) - 3 + 2

⇥ⓘ Subtracting a negative number is adding it, two minus signs make a plus:

- (-1)

1

■Start from

-3

and move forward

2

on the number line, getting closer to zero:

- 3 + 2

-1

⇥

1 - 1

⇥ⓘ Start from

1

and move back

1

on the number line, down to zero:

1 - 1

0

⇥

0

⇥Root found:

λ = -1

⇥1.2) Perform polynomial long division of

(- λ^{3} + 3 λ + 2)

(λ + 1)

⇥

Expression in progress	Divide leading terms	Quotient	Quotient × divisor	Remainder after subtracting
$- λ^{3} + 3 λ + 2$	$- \frac{λ^{3}}{λ}$	$- λ^{2}$	$- λ^{2} \cdot (λ + 1) = - λ^{3} - λ^{2}$	$(- λ^{3} + 3 λ + 2) - (- λ^{3} - λ^{2}) = λ^{2} + 3 λ + 2$
$λ^{2} + 3 λ + 2$	$\frac{λ^{2}}{λ}$	$λ$	$λ \cdot (λ + 1) = λ^{2} + λ$	$(λ^{2} + 3 λ + 2) - (λ^{2} + λ) = 2 λ + 2$
$2 λ + 2$	$\frac{2 λ}{λ}$	$2$	$2 \cdot (λ + 1) = 2 λ + 2$	$(2 λ + 2) - (2 λ + 2) = 0$

⇥So the result of the euclidean division is:

- λ^{2} + λ + 2

⇥1.3) Let's factor in

ℝ

(- λ^{2} + λ + 2)

⇥Given the quadratic form

a λ^{2} + b λ + c

, you can calculate the discriminant:

Δ = b^{2} - 4 a c

. Here

a = -1

b = 1

and

c = 2

⇥

Δ = 1 - 4 \cdot (-1) \cdot 2

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

-4 \cdot (-1)

4

⇥

Δ = 1 + 4 \cdot 2

⇥ⓘ Multiplications are done before additions and subtractions:

4 \cdot 2

8

⇥

Δ = 1 + 8

⇥ⓘ Start from

1

and move forward

8

on the number line:

1 + 8

9

⇥

Δ = 9

⇥Since

Δ > 0

, the equation has two roots:

⇥The quadratic formula gives

λ_{1}

\frac{- b - \sqrt{Δ}}{2 a}

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

⇥

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

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

\sqrt{9}

3

;

3^{2}

9

■A single minus sign does not cancel out, the product of a negative number by a positive number is negative:

2 \cdot (-1)

-2

⇥

λ_{1} = \frac{- 1 - 3}{-2}

⇥ⓘ The sign of a fraction goes in front of the bar, whether it comes from the numerator or the denominator:

\frac{- 1 - 3}{-2}

- (\frac{- 1 - 3}{2})

⇥

λ_{1} = - (\frac{- 1 - 3}{2})

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

- 1 - 3

-4

⇥

λ_{1} = - (\frac{-4}{2})

⇥ⓘ Two minus signs cancel out:

- (\frac{-4}{2})

\frac{4}{2}

⇥

λ_{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

⇥

λ_{1} = 2

⇥The quadratic formula gives

λ_{2}

\frac{- b + \sqrt{Δ}}{2 a}

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

⇥

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

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

\sqrt{9}

3

;

3^{2}

9

■A single minus sign does not cancel out, the product of a negative number by a positive number is negative:

2 \cdot (-1)

-2

⇥

λ_{2} = \frac{- 1 + 3}{-2}

⇥ⓘ The sign of a fraction goes in front of the bar, whether it comes from the numerator or the denominator:

\frac{- 1 + 3}{-2}

- (\frac{- 1 + 3}{2})

⇥

λ_{2} = - (\frac{- 1 + 3}{2})

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

- 1 + 3

2

⇥

λ_{2} = - \frac{2}{2}

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

2

, their greatest common divisor:

- \frac{2}{2}

-1

⇥

λ_{2} = -1

▷The factor

(- λ^{2} + λ + 2)

factors as:

- (λ + 1) \cdot (λ - 2)

2) Let's replace each factor by its factorization:

(λ + 1) \cdot (- (λ + 1) \cdot (λ - 2))

ⓘ The minus sign is placed in front of the product:

(λ + 1) \cdot (- (λ + 1)) \cdot (λ - 2)

- (λ + 1) \cdot (λ + 1) \cdot (λ - 2)

ⅰ

- λ^{2} + λ + 2

- (λ + 1) \cdot (λ - 2)

- (λ + 1) \cdot (λ + 1) \cdot (λ - 2)

▶Factored expression:

- {(λ + 1)}^{2} \cdot (λ - 2)