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In algebra, the Greatest Common Factor (GCF) is extremely useful. It refers to the largest number or term that can be divided evenly into each term of a polynomial.
To find the GCF, you first look at the coefficients (numbers in front of variables) of the terms. For instance, in the polynomial given, the coefficients are 65 and 35.
By identifying the highest number that can divide both 65 and 35, you find the GCF for the coefficients which is 5.
Next, examine the variable parts. You will look at the exponents of the variables and choose the smallest exponent shared by all terms. Here, we have exponents 10 and 6 for the variable y, and the smallest is 6.
Thus, the GCF of the polynomial 65y^{10} + 35y^{6} is 5y^6.
Breaking down problems into smaller parts like this makes it easier to solve them.

The factored form of a polynomial is a way of expressing it as the product of its factors. This is helpful in simplifying expressions and solving equations.
To factor a polynomial means to break it down into simpler 'building blocks' that, when multiplied together, give back the original polynomial.
In the example: 65y^{10} + 35y^{6}, we already identified the GCF as 5y^6.
By factoring out 5y^6 from each term, we simplify:
65y^{10} + 35y^{6} = 5y^6 (13y^4) + 5y^6 (7).
When we combine terms: 65y^{10} + 35y^{6} = 5y^6 (13y^4 + 7).
This new expression 5y^6 (13y^4 + 7) is the factored form.
The polynomial is now simpler and reveals more about its structure, making further algebraic manipulations more straightforward.

Algebra is the branch of mathematics dealing with variables and the rules for manipulating these variables. It forms the foundation for more advanced topics in math and science.
Working with algebra involves a few key concepts like understanding variables, expressions, and solving equations.
For example, in this exercise, we use basic algebraic techniques to simplify a polynomial by factoring it.
Polynomials are expressions with multiple terms often involving variables raised to various powers.
Knowing how to factor these expressions is crucial because it allows you to simplify complex equations and solve for unknown variables more easily.
Practicing these techniques helps build a strong foundation in algebra, enabling students to tackle more complicated problems with confidence.