High School

Cecile used the [tex]$X$[/tex] method to factor [tex]$16x^6 - 9$[/tex].

1. Rewrite as: [tex]$16x^6 + 0x - 9$[/tex]

2. Apply the [tex]$X$[/tex] method:

[tex]$16x^6 + 12x^3 - 12x^3 - 9$[/tex]

3. Factor by grouping:

[tex]$4x^3(4x^3 + 3) + (-3)(4x^3 + 3)$[/tex]

4. Final factored form:

[tex]$(4x^3 + 3)(4x^3 - 3)$[/tex]

Answer :

To factor the polynomial [tex]\(16x^6 - 9\)[/tex] using the steps Cecile outlined, let's walk through the process step by step:

### Step 1: Recognize the form

The expression [tex]\(16x^6 - 9\)[/tex] is a difference of squares. We can write it as:

[tex]\[
(4x^3)^2 - 3^2
\][/tex]

### Step 2: Apply the Difference of Squares Formula

The difference of squares formula is:

[tex]\[
a^2 - b^2 = (a + b)(a - b)
\][/tex]

In this case, [tex]\(a = 4x^3\)[/tex] and [tex]\(b = 3\)[/tex].

### Step 3: Substitute and Factor

Using the formula, we get:

[tex]\[
(4x^3)^2 - 3^2 = (4x^3 + 3)(4x^3 - 3)
\][/tex]

### Conclusion

The factored form of the polynomial [tex]\(16x^6 - 9\)[/tex] is:

[tex]\[
(4x^3 + 3)(4x^3 - 3)
\][/tex]

This is the final answer in its factored form.

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