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Polynomial factoring is a critical skill in algebra that helps simplify equations and solve for variables. Polynomials are mathematical expressions involving a sum of powers in one or more variables multiplied by coefficients. For instance, in the exercise, the given polynomial is expressed as multiple terms with varying powers of the variables t and s: \[10t^{3}-2t^{2}s^{2}-5ts+s^{3}\]. By factoring polynomials, you break them down into simpler polynomial factors that, when multiplied together, give back the original polynomial. This approach simplifies solving equations or finding roots. To factor by grouping, you look for ways to rearrange the polynomial into groups that share common factors, then factor those out step by step.

The Greatest Common Factor (GCF) is the largest factor or divisor that divides two or more numbers or terms without leaving a remainder. When factoring polynomials by grouping, identifying the GCF within each group of terms is essential. In our exercise, after grouping the polynomial into two pairs:\[(10t^{3} - 2t^{2}s^{2}) + (-5ts + s^{3})\] it's easy to spot that in the first group, the common factor is \(2t^{2}\), and in the second group, the common factor is \(s\). Factoring these out simplifies each group to: \[2t^{2}(5t - s^{2}) - s(5t - s^{2})\].Notice that both factored groups now share a common binomial \((5t - s^{2})\). This makes it easier to factor the entire polynomial.

After factoring out the Greatest Common Factor from each group, the next step is to look for common binomial factors. These are expressions of the form \((a + b)\) or \((a - b)\) that appear in multiple groups. In our example:\[2t^{2}(5t - s^{2}) - s(5t - s^{2})\], the common binomial factor present in both groups is \((5t - s^{2})\). By factoring this binomial out, you simplify the polynomial even more: \[(5t - s^{2})(2t^{2} - s)\].Now, the original polynomial is factored into the product of two binomials, which makes solving equations or simplifying expressions involving the polynomial more straightforward and manageable.