(x^2-4)(x^2+6x+9) Factored Completely

2 min read Jun 17, 2024
(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.

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