factor the following polynomial completely
2b^3-18b
To factor the given polynomial, 2b^3 - 18b, completely, we first look for the greatest common factor (GCF) in both terms.
The GCF of 2b^3 and -18b is 2b since 2 is the largest number that divides evenly into both 2 and 18, and b is the highest power of b common to both terms.
First, we factor out the GCF:
2b^3 - 18b = 2b(b^2 - 9)
We can observe that the expression inside the parentheses (b^2 - 9) is a difference of squares. This can be factored further because a difference of squares follows the pattern:
a^2 - b^2 = (a + b)(a - b).
In our case, b^2 - 9 can be written as:
b^2 - 3^2
Now we can factor the difference of squares:
b^2 - 3^2 = (b + 3)(b - 3)
Thus, the complete factorization of the polynomial 2b^3 - 18b is:
2b(b + 3)(b - 3).