HOW DO YOU FACTOR A^3-27. PLEASE EXPLAIN IN DETAIL
it follows the difference of cubes pattern
A^3 - B^3 = (A-B)(A^2 + AB + b^2)
so (A-3)(.......)
tell me what you got.
We have a difference of cubes.
Did you apply the formula given by Reiny?
If you did, there should be no more doubts.
A^3 - 27 = difference of cubes.
First, write 27 as an exponent.
What number when multiplied by itself 3 times will yield 27?
How about 3?
So, 3 x 3 x 3 = 27, right? This can be written 3^3.
We now have this:
A^3 - 3^3
Next, we apply the rule given to you by Reiny.
A^3 - B^3 = (A-B)(A^2 + AB + b^2)
Here, A = A and B = 3.
We plug and chug.
A^3 - 3^3 = (A - 3)(A^2 + 3A + 3^2)
A^3 - 3^3 = (A - 3)(A^2 + 3A + 9)
Done!
To factor the expression a^3 - 27, we need to recognize that it is in the form of a difference of cubes, which can be factored using a specific formula:
a^3 - b^3 = (a - b)(a^2 + ab + b^2)
In this case, a^3 - 27 can be rewritten as:
a^3 - 3^3
Comparing this with the formula, we can see that a is equal to 'a' and b is equal to 3.
Now, we can plug the values of 'a' and 'b' into the formula:
(a - b)(a^2 + ab + b^2)
Substituting the values, we get:
(a - 3)(a^2 + 3a + 9)
Therefore, the factored form of a^3 - 27 is (a - 3)(a^2 + 3a + 9).
Remember, recognizing patterns and formulas is crucial when factoring polynomials. In this case, understanding the difference of cubes formula allowed us to identify the factors and simplify the expression.