Factor the following polynomial completely.

Question

4 x Superscript 4 Baseline minus 48 x squared plus 144
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Part 1
4 x Superscript 4 Baseline minus 48 x squared plus 144equals
  
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Answer 1

To factor the polynomial \(4x^4 - 48x^2 + 144\), we can start by looking for a common factor in all the terms.

1. Factor out the Greatest Common Factor (GCF): The GCF of \(4x^4\), \(-48x^2\), and \(144\) is \(4\). So we can factor that out first:

   \[ 4(x^4 - 12x^2 + 36) \]

2. Substitute: Next, let's make the substitution \(y = x^2\). Thus, the expression becomes:

   \[ 4(y^2 - 12y + 36) \]

3. Factor the quadratic: Now we need to factor the quadratic \(y^2 - 12y + 36\). We look for two numbers that multiply to \(36\) and add to \(-12\). These numbers are \(-6\) and \(-6\).

   Thus, we can factor it as:

   \[ y^2 - 12y + 36 = (y - 6)(y - 6) = (y - 6)^2 \]

4. Substitute back: Now replace \(y\) with \(x^2\):

   \[ (x^2 - 6)^2 \]

5. Combine everything: Therefore, the expression can be written as:

   \[ 4(x^2 - 6)^2 \]

6. Final representation: The fully factored form of the polynomial is:

   \[ 4(x^2 - 6)^2 \]

So, the final answer is:

\[ 4(x^2 - 6)^2 \]