Factoring Quadratics: GCF and Prime Steps

Welcome to our lesson on finding the Greatest Common Factor in algebraic expressions!Let's work with this example: six x squared plus twelve x plus eighteen.First, we need to identify each term in the expression.Now, let's list out all the factors for each term, including variables and their exponents.Looking at our lists, we can identify the factors that are common to all terms.The numbers that divide evenly into six x squared, twelve x, and eighteen are one, two, three, and six.Notice that while x appears in some terms, it's not common to all terms, so it won't be part of our Greatest Common Factor.Therefore, the Greatest Common Factor of our expression is six, as it's the largest number that divides evenly into all terms.Let's look at another example to reinforce our understanding.In this expression, we have three terms with both x and y variables.Looking at the terms, we can find that fifteen is the greatest common factor among the numbers, x to the first power is common to all terms, and y to the first power is also common to all terms.Therefore, the Greatest Common Factor for this expression is fifteen x y.Now that we can find the Greatest Common Factor, we're ready for the next step in factoring.Now that we have our GCF of 6, let's factor it out of each term.We'll divide each term by 6 to separate our GCF from the remaining factors.Let's look at each term individually.Six x squared divided by six gives us x squared.Twelve x divided by six gives us two x.And eighteen divided by six gives us three.Now we can write our expression in factored form, with six as our GCF outside the parentheses.To verify our factoring is correct, we can distribute the six back through each term.When we multiply six by each term inside the parentheses, we get back our original expression.After factoring out the GCF, we need to analyze the remaining coefficients through prime factorization.Let's look at our example: x squared plus seven x plus twelve. The coefficient of the first term is 1, which is already prime.For the last term, twelve, we'll create a factor tree to find its prime factorization.Twelve breaks down into two times two times three.Using these prime factors, we can identify possible factor pairs that could give us our trinomial.Let's verify one pair using the FOIL method. We'll try x plus three times x plus four.Let's review the key points about prime factorization in polynomial factoring.Thanks for learning about prime factorization in polynomial factoring with Spark.E!