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When factoring expressions, the greatest common factor (GCF) is a key concept. It represents the largest number that divides all terms in the expression without leaving a remainder. Finding the GCF is the first step in simplifying an expression and breaking it down into its factors. Consider the expression given: \[2t^2 - 24ts + 70s^2\]Here, you need to identify the GCF for the terms \(2t^2\), \(-24ts\), and \(70s^2\). Each term can be divided by 2, which is the GCF. By factoring out 2, you simplify the expression and make further factorization easier. After dividing each term by 2, the expression becomes:2(t^2 - 12ts + 35s^2).
Finding the GCF isn't just a step in the process—it's fundamental in ensuring that the expression is completely factored in later steps.

Factoring polynomials involves breaking the expression into simpler terms that, when multiplied together, produce the original polynomial. Once you've factored out the GCF, the next focus is on the remaining polynomial:\[t^2 - 12ts + 35s^2\]The objective is to express it as a product of two binomials. To do this, you need to find two numbers that add up to the coefficient of the middle term (-12) and multiply to the constant term (+35). These numbers are -5 and -7.
Express the polynomial as the product of two factors:(t - 5s)(t - 7s).
This process involves understanding how numbers interact in addition and multiplication, emphasizing the connections between coefficients and constant terms within the polynomial. Such insights can simplify complex problems into manageable tasks.

Quadratic equations form when you set a quadratic expression equal to zero. They have the standard form:\[ax^2 + bx + c = 0\].
The quadratic equation differs slightly from our original problem, but the underlying principles of factorization remain similar. By converting the expression \[2(t-5s)(t-7s)\]into a possible scenario where it equals zero, you can solve for variable values that satisfy the equation, illustrating how expressions factor into workable solutions.
Here’s the breakdown:

• Setting each factor equal to zero gives solutions for possible variable values.
• Understanding that each term in the equation interacts through multiplication.

By focusing on these solutions, you uncover the usefulness of fully factored quadratic expressions. Factorization simplifies complex equations, revealing roots or solutions that fit specific conditions and making algebraic challenges more approachable.