Question

Factor the following polynomial completely.
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
  
enter your response here ​(Type your answer in factored​ form.)

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:

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