(x%5E2%2B6x%2B9)-factored-completely.jpeg "(x^2-4)(x^2+6x+9) Factored Completely")
Factoring the Expression (x^2 - 4)(x^2 + 6x + 9)
This expression represents the product of two binomials. To factor it completely, we need to factor each binomial individually.
Factoring (x^2 - 4)
This binomial is a difference of squares, where:
- x^2 is the square of x
- 4 is the square of 2
The difference of squares pattern is: a^2 - b^2 = (a + b)(a - b)
Applying this pattern to our binomial:
- a = x
- b = 2
Therefore, (x^2 - 4) = (x + 2)(x - 2)
Factoring (x^2 + 6x + 9)
This binomial is a perfect square trinomial, where:
- The first term (x^2) is the square of x
- The last term (9) is the square of 3
- The middle term (6x) is twice the product of x and 3 (2 * x * 3 = 6x)
The perfect square trinomial pattern is: a^2 + 2ab + b^2 = (a + b)^2
Applying this pattern to our binomial:
- a = x
- b = 3
Therefore, (x^2 + 6x + 9) = (x + 3)^2
Combining the Factors
Now that we have factored both binomials, we can substitute them back into the original expression:
(x^2 - 4)(x^2 + 6x + 9) = (x + 2)(x - 2)(x + 3)^2
This is the completely factored form of the expression.