Question

factor completely.
5
$x^4 - 81$
6
$36x^2 + 84x + 49$
7
$6x^2 - 24$
8
$242x^2 + 44x + 2$

Explanation:

Problem 5: Factor \( x^4 - 81 \)

Step 1: Recognize as difference of squares

\( x^4 - 81 = (x^2)^2 - 9^2 \)
Using the difference of squares formula \( a^2 - b^2 = (a - b)(a + b) \), we get:
\( (x^2 - 9)(x^2 + 9) \)

Step 2: Factor \( x^2 - 9 \) further

\( x^2 - 9 = x^2 - 3^2 \), which is also a difference of squares.
Applying the formula again:
\( (x - 3)(x + 3) \)
So the completely factored form is \( (x - 3)(x + 3)(x^2 + 9) \)

Step 1: Recognize as a perfect square trinomial

Notice that \( 36x^2 = (6x)^2 \), \( 49 = 7^2 \), and \( 84x = 2 \times 6x \times 7 \)
The perfect square trinomial formula is \( a^2 + 2ab + b^2 = (a + b)^2 \)
Here, \( a = 6x \) and \( b = 7 \)

Step 2: Apply the formula

\( 36x^2 + 84x + 49 = (6x + 7)^2 \)

Step 1: Factor out the greatest common factor (GCF)

The GCF of \( 6x^2 \) and \( 24 \) is \( 6 \)
\( 6x^2 - 24 = 6(x^2 - 4) \)

Step 2: Factor \( x^2 - 4 \)

\( x^2 - 4 = x^2 - 2^2 \), which is a difference of squares.
Using the formula \( a^2 - b^2 = (a - b)(a + b) \):
\( x^2 - 4 = (x - 2)(x + 2) \)
So the completely factored form is \( 6(x - 2)(x + 2) \)

Answer:

\( (x - 3)(x + 3)(x^2 + 9) \)

Problem 6: Factor \( 36x^2 + 84x + 49 \)