Answer :
[tex]20x^3 - 45[/tex] is completely factored as [tex]5(2x - 3)(2x + 3)[/tex].
To factor the expression [tex]20x^3 - 45[/tex] completely:
Identify the Greatest Common Factor (GCF):
First, we need to find the GCF of the coefficients 20 and 45.
The factors of 20 are: 1, 2, 4, 5, 10, 20
The factors of 45 are: 1, 3, 5, 9, 15, 45
The GCF is 5.
Factor out the GCF:
Now, we can factor out 5 from the expression.
[tex]20x^3 - 45 = 5(4x^3 - 9)[/tex]
Factor the difference of squares:
Notice that [tex]4x^3 - 9[/tex] can be recognized as a difference of squares since [tex]4x^3 = (2x)^{2}[/tex] and [tex]9 = 3^2[/tex].
Thus, we can rewrite it as:
[tex]4x^3 - 9 = (2x)^{2} - 3^{2}[/tex]
This follows the difference of squares factoring pattern:
[tex]a^2 - b^2 = (a - b)(a + b)[/tex].
Here, let [tex]a = 2x[/tex] and [tex]b = 3[/tex]:
[tex]4x^3 - 9 = (2x - 3)(2x + 3)[/tex]
Combine everything:
Now, substituting back into our expression gives:
[tex]20x^3 - 45 = 5(2x - 3)(2x + 3)[/tex]
Final Factored Form:
Therefore, the complete factored form of the expression is:
[tex]20x^3 - 45 = 5(2x - 3)(2x + 3)[/tex]
