91Ó°ÊÓ

Grouping terms is a helpful technique in factoring polynomials. It involves rearranging and grouping terms within the polynomial to make it easier to identify common factors. In our exercise, we started with the polynomial: \[ 5x^{3} - 3x^{2} + 5x - 3 \]To simplify the process, group the terms in pairs: \[ (5x^{3} - 3x^{2}) + (5x - 3) \]Notice how we grouped the terms such that similar expressions are together. This step sets up the polynomial for easy factoring in subsequent steps. Remember, grouping terms effectively can significantly simplify the problem, making it less complex to solve. This technique is particularly useful when you encounter polynomials with four or more terms.

After grouping the terms, the next step is to factor out the common factor from each group. A common factor is a number or expression that divides each term of the given polynomial without leaving a remainder. In our exercise, we have the grouped terms: \[ (5x^{3} - 3x^{2}) + (5x - 3) \]Now, extract the greatest common factor from each group. For the first group, \(5x^{3} - 3x^{2}\), the common factor is \(x^{2}\):\[ x^{2}(5x - 3) \]In the second group, \(5x - 3\), there’s no common factor, so it remains as is: \[ (5x - 3) \]Now, combine these results: \[ x^{2}(5x - 3) + (5x - 3) \]By factoring out the common factor at this stage, we make it easier to perform the final factoring step. This process highlights the importance of identifying common factors to simplify polynomials.

The final goal of factoring is to express the polynomial in its simplest, factored form. In our exercise, after factoring out the common factor in each group, we have: \[ x^{2}(5x - 3) + (5x - 3) \] Notice that \(5x - 3\) is a common binomial factor in both terms. So, we can factor out \(5x - 3\): \[ (5x - 3)(x^{2} + 1) \]This is the factored form of the polynomial. The factored form is particularly useful because it simplifies complex expressions, making it easier to analyze and solve equations involving polynomials. It reveals the roots and intercepts more clearly, which can be very helpful in graphing or solving polynomial equations. Mastering the skill of factoring will greatly enhance your mathematical toolkit, allowing you to tackle a wide range of algebraic problems more efficiently.