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

1) Let's factor in

ℝ

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

For

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

, we recognize a difference of two squares:

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

. Apply

a^{2} - b^{2} = (a - b) \cdot (a + b)

, with

a = x^{2}

and

b = 2 x

You can factorize the expression:

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

into:

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

2) Let's factor in

ℝ

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

⇥2.1) Let's factor in

ℝ

(x^{2} - 2 x)

⇥You can write the expression in factored form:

x \cdot (x - 2)

▷The factor

(x^{2} - 2 x)

factors as:

x \cdot (x - 2)

⇥2.2) Let's factor in

ℝ

(x^{2} + 2 x)

⇥This expression can be factorized into:

x \cdot (x + 2)

▷The factor

(x^{2} + 2 x)

factors as:

x \cdot (x + 2)

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

x \cdot (x - 2) \cdot x \cdot (x + 2)

ⓘ Multiplying powers with the same base adds the exponents:

x \cdot x

x^{2}

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

ⓘ Factor out the common factor

x^{2}

▶Factored expression:

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