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In algebra, the Greatest Common Factor (GCF) is a key tool for simplifying expressions. When you have a polynomial, just like the expression \(-3x^2 + 30x - 75\), finding the GCF can make the process easier before addressing any deeper factoring issues. To find the GCF, examine all the coefficients in your polynomial.
Here, the coefficients are \(-3\), \(30\), and \(-75\). Notice that they all share a common factor of \(3\). Since \(-3\) itself is a factor of each, and because the leading term is negative, \(-3\) is considered the greatest common factor.
Once identified, you factor out \(-3\) from each term. This consolidation significantly reduces complexity and sets a foundation for further simplification.
By factoring it out, the polynomial simplifies to \(-3(x^2 - 10x + 25)\). This essential step ensures the problem is easier to manage as you progress.

Trinomial factoring turns a three-term polynomial into a simpler expression of two binomials or sometimes a squared binomial. In this case, the expression has already been reduced by factoring out the GCF, leaving us with \(x^2 - 10x + 25\).
To factor a trinomial like \(x^2 - 10x + 25\), examine its form and characteristics:

• The coefficient of \(x^2\) is \(1\), making this a straightforward case of trinomial factoring.
• The middle term, \(-10x\), and the constant term, \(25\), suggest potential special patterns, such as perfect squares.

In this case, it's important to see that the trinomial is a perfect square trinomial (as \((-5)\times(-5) = 25\) and \(2\times(-5) = -10\)).
This recognition greatly simplifies the process by allowing us to rewrite the trinomial as \((x - 5)^2\).
Thus, a complex polynomial is now neatly expressed as a binomial square through effective trinomial factoring.

Perfect square trinomials have a specific structure that makes them stand out within polynomials. They take the form \(a^2 \pm 2ab + b^2\), allowing them to be written as \((a \pm b)^2\).
In our example, the phrase \(x^2 - 10x + 25\) fits this pattern perfectly:

• \(x^2\) is \(a^2\), where \(a = x\)
• \(25\) is \(b^2\), where \(b = 5\)
• The middle term, \(-10x\), matches \(-2ab\).

The symmetry and balance in perfect square trinomials allow for quick conversion into a squared binomial. This is valuable for simplifying or solving quadratic expressions.
Recognizing perfect square trinomials not only streamlines the factoring process, but also provides insight into their graphical representation, as they are always non-negative and touch the x-axis at one point, making them perfect squares visually as well.