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When working with polynomials, one of the first things to check is the greatest common factor (GCF). This involves finding the largest number and variable expression that divides each term in the polynomial without leaving a remainder.
For example, in the polynomial \(-15a^2 - 40a\), we begin by examining the coefficients \(-15\) and \(-40\).
Here is a simple guide to finding the GCF:

• List the factors of each coefficient. For \(-15\), they are \[1, 3, 5, 15\]. For \(-40\), they are \[1, 2, 4, 5, 8, 10, 20, 40\]. The largest common factor is \(-5\).
• Look at the variables. For \(a^2\) and \(a\), the smallest power is \(a\).

Therefore, combining the results, the GCF of \(-15a^2\) and \(-40a\) is \(-5a\).
Factoring out the GCF simplifies the polynomial, making it easier to work with in subsequent steps.

After identifying the greatest common factor (GCF), the next step is to factor it out from the polynomial by performing polynomial division.
This involves dividing each term of the polynomial by the GCF. Let's continue with our example \(-15a^2 - 40a\):

• Divide \(-15a^2\) by the GCF \(-5a\): \( -15a^2 \,\ -5a = 3a \).
• Divide \(-40a\) by the GCF \(-5a\): \(-40a \,\ -5a = 8 \).

Putting it all together, we rewrite the polynomial as:
\(-15a^2 - 40a = -5a(3a + 8) \).
This step-by-step division helps in simplifying the polynomial, making it more straightforward and manageable for further operations or solving equations.

Factoring polynomials is an essential technique in algebra that helps simplify expressions and solve equations. Factoring involves rewriting the polynomial as a product of its factors.
One common approach is to factor out the greatest common factor (GCF) first, as we did in the previous sections. This step simplifies the polynomial and can reveal patterns or further factorable expressions inside the parentheses.
Here are a few key points about the factoring technique:

• Always start by identifying and factoring out the GCF.
• After removing the GCF, check if the resulting polynomial can be factored further (e.g., by recognizing common patterns like difference of squares or trinomials).

In our example, we identified \(-5a\) as the GCF and factored the polynomial \(-15a^2 - 40a\) to get \(-5a(3a + 8)\). This final expression is much simpler to work with, whether for solving equations or other algebraic manipulations.
Mastering this sequence of GCF identification, polynomial division, and further factoring techniques will make algebra problems more approachable and easier to solve.