How do you factor the difference of two squares? How do you factor the perfect square trinomial? How do you factor the sum and difference of two cubes? Which of these three makes the most sense to you? Explain why.
Difference of two squares:
Example: x^2 - 9 factors to (x+3)(x-3)
Perfect squares:
x^2 + 6x + 9 = (x+3)(x+3)
x^2 -6x + 9 = (x-3)(x-3)
Cubes:
x^3 - 27 factors to
(x-3)(x^2 +3x +9)
x^3 + 27 factors to
(x+3)(x^2 -3x +9)
Can you see the pattern?
To factor the difference of two squares, you need to identify an expression in the form of "a^2 - b^2." This can be factored as "(a + b)(a - b)." For example, if you have the expression x^2 - 9, you can factor it as (x + 3)(x - 3).
To factor a perfect square trinomial, you need to find an expression in the form of "a^2 + 2ab + b^2" or "a^2 - 2ab + b^2." This can be factored as "(a + b)^2" or "(a - b)^2." For example, if you have the expression x^2 + 6x + 9, you can factor it as (x + 3)^2.
To factor the sum or difference of two cubes, you need to recognize an expression in the forms "a^3 + b^3" or "a^3 - b^3." These can be factored using the formulas "(a + b)(a^2 - ab + b^2)" or "(a - b)(a^2 + ab + b^2)." For example, if you have the expression x^3 + 8, you can factor it as (x + 2)(x^2 - 2x + 4).
Out of these three, the most logical and straightforward to me is factoring the difference of two squares. This is because the pattern "a^2 - b^2 = (a + b)(a - b)" is easy to remember and apply. Additionally, it is a commonly used factorization method in many mathematical situations, so it is an important skill to have.