To factor the polynomial \( 48x^6 - 3x^2 \), we can start by factoring out the greatest common factor (GCF).
- Identify the GCF:
- The coefficients are 48 and -3. The GCF of 48 and 3 is 3.
- The variable part has \( x^6 \) and \( x^2 \). The GCF for the variables is \( x^2 \).
So, the GCF of the entire polynomial is \( 3x^2 \).
-
Factor out the GCF from the polynomial: \[ 48x^6 - 3x^2 = 3x^2(16x^4 - 1) \]
-
Now, we need to factor \( 16x^4 - 1 \). This expression is a difference of squares: \[ 16x^4 - 1 = (4x^2)^2 - 1^2 = (4x^2 - 1)(4x^2 + 1) \]
-
\( 4x^2 - 1 \) can be further factored, as it is also a difference of squares: \[ 4x^2 - 1 = (2x - 1)(2x + 1) \]
-
The term \( 4x^2 + 1 \) cannot be factored over the real numbers, as it does not have real roots.
Putting it all together, the completely factored form of the polynomial \( 48x^6 - 3x^2 \) is: \[ 3x^2(2x - 1)(2x + 1)(4x^2 + 1) \]
Thus, the final answer is: \[ \boxed{3x^2(2x - 1)(2x + 1)(4x^2 + 1)} \]
1 answer