91Ó°ÊÓ

When you see the term ‘difference of squares,’ it refers to a specific kind of polynomial. A difference of squares is structured as \[ a^2 - b^2 \].
This can be factored using the formula: \[ a^2 - b^2 = (a - b)(a + b) \].
The problem given, \[ 4r^2 - 12r + 9 - s^2 \], can be fit into this pattern after recognizing its components.
Notice how the polynomial can be rewritten: \[ (2r - 3)^2 - s^2 \].
Here, \[ (2r - 3)^2 \] acts as your \[ a^2 \] and \[ s^2 \] is your \[ b^2 \].
Using the difference of squares formula results in: \[ (2r - 3 - s)(2r - 3 + s) \].
This method is invaluable for simplifying polynomials that match this pattern.

A perfect square trinomial is a trinomial that factors into a binomial squared. For example, \[ 4r^2 - 12r + 9 \] is a perfect square trinomial.
It can be expressed as \[ (2r - 3)^2 \].
Recognizing a perfect square trinomial involves identifying patterns in the coefficients:
- The first term should be a squared term (like \[ 4r^2 \]).
- The last term should be another squared term (here, \[ 9 \]).
- The middle term should be twice the product of the square roots of the first and third terms (which gives us \[ 12r \]).
In simpler terms, if you redistribute \[ (2r - 3) \], you get: \[ (2r - 3)(2r - 3) = 4r^2 - 12r + 9 \].
Breaking down the trinomial helps in identifying it and hence factoring it correctly.

Factoring polynomials involves various formulas and strategies. Key formulas include:

• Difference of Squares: \[ a^2 - b^2 = (a - b)(a + b) \]
• Perfect Square Trinomial: \[ a^2 + 2ab + b^2 = (a + b)^2 \] and \[ a^2 - 2ab + b^2 = (a - b)^2 \]
• Sum of Cubes: \[ a^3 + b^3 = (a + b)(a^2 - ab + b^2) \]
• Difference of Cubes: \[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \]

In this exercise, we focused on combining the difference of squares and perfect square trinomial formulas.
Recognizing the structure of \[ (2r - 3)^2 - s^2 \] allows application of the difference of squares formula, yielding: \[ (2r - 3 - s)(2r - 3 + s) \].
Understanding these core formulas is essential for simplifying and solving polynomial equations efficiently.