factor x^3-3x^2-4x — step-by-step solution

Let's factor in

ℝ

x^{3} - 3 x^{2} - 4 x

This expression can be factorized into:

x \cdot (x^{2} - 3 x - 4)

Let's factorize in

ℝ

(x^{2} - 3 x - 4)

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 = -3

and

c = -4

Δ = {(-3)}^{2} - 4 \cdot 1 \cdot (-4)

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

{(-3)}^{2}

(-3) \cdot (-3)

9

Δ = 9 - 4 \cdot (-4)

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

- 4 \cdot (-4)

16

Δ = 9 + 16

ⓘ Start from

9

and move forward

16

on the number line:

9 + 16

25

Δ = 25

Δ > 0

, the equation has two roots:

The quadratic formula gives

x_{1}

\frac{- b - \sqrt{Δ}}{2 a}

\frac{- (-3) - \sqrt{25}}{2 \cdot 1}

x_{1} = \frac{3 - \sqrt{25}}{2}

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

\sqrt{25}

5

;

5^{2}

25

x_{1} = \frac{3 - 5}{2}

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

3 - 5

-2

x_{1} = - \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_{1} = -1

The quadratic formula gives

x_{2}

\frac{- b + \sqrt{Δ}}{2 a}

\frac{- (-3) + \sqrt{25}}{2 \cdot 1}

x_{2} = \frac{3 + \sqrt{25}}{2}

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

\sqrt{25}

5

;

5^{2}

25

x_{2} = \frac{3 + 5}{2}

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

3 + 5

8

x_{2} = \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_{2} = 4

▷The factor

(x^{2} - 3 x - 4)

factors as:

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

ⓘ Factor out the common factor

x

▶Factored expression:

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