domain of (x^3+7x^2+12x)/(x^3+x^2-2x)

1) Let's find the domain of

\frac{x^{3} + 7 x^{2} + 12 x}{x^{3} + x^{2} - 2 x}

For the expression

\frac{x^{3} + 7 x^{2} + 12 x}{x^{3} + x^{2} - 2 x}

to exist, you must check:

x^{3} + x^{2} - 2 x \neq 0

You can write the expression:

x^{3} + x^{2} - 2 x

in factored form:

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

⇥1.1) Let's solve in

ℝ

x \neq 0

⇥Solution obtained:

x \neq 0

⇥1.2) Let's solve in

ℝ

x^{2} + x - 2 \neq 0

⇥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 = -2

⇥

Δ = 1 - 4 \cdot 1 \cdot (-2)

⇥

Δ = 1 - 4 \cdot (-2)

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

- 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

x_{1}

\frac{- b - \sqrt{Δ}}{2 a}

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

⇥

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

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

\sqrt{9}

3

;

3^{2}

9

⇥

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

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

- 1 - 3

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

⇥

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

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

\sqrt{9}

3

;

3^{2}

9

⇥

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

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

- 1 + 3

2

⇥

x_{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

⇥

x_{2} = 1

⇥Solution found:

x \in ℝ \ {-2, 1}

▶Domain of definition:

x \in ℝ \ {-2, 0, 1}