Factoring Polynomials Step by Step with Animations

Welcome to our lesson on identifying terms and finding common factors in polynomials!Let's examine this polynomial: six x squared y plus twelve x y squared plus three x y.First, let's identify each term in our polynomial.Let's break down each term into its coefficient and variables.Now, let's find the greatest common factor. We'll look at both coefficients and variables.Looking at the coefficients: six, twelve, and three. The greatest common factor is three.For variables, we can see that x y appears in every term, making it our common variable factor.Therefore, three x y is our greatest common factor. Let's factor it out.We can verify this by dividing each term by three x y.We can verify our factorization by distributing three x y back to each term.After removing common factors, we need to look for special patterns that can help us factor expressions.The difference of squares pattern occurs when we subtract two squared terms. For example, x squared minus sixteen can be factored as x plus four times x minus four.Perfect square trinomials are expressions that can be written as a binomial squared. They have a specific pattern: a squared term, plus or minus twice the product of the terms, plus the second term squared.The sum or difference of cubes follows a special pattern. For example, x cubed minus eight factors as x minus two times x squared plus two x plus four.Let's practice identifying these patterns with some examples. Here's a difference of squares: four x squared minus twenty-five.And here's a perfect square trinomial: x squared minus ten x plus twenty-five.Watch how we can transform this expression step by step to recognize the perfect square pattern.For polynomials that don't fit standard patterns, we can use the grouping method.First, we group terms that might have common factors.Next, factor each group separately to find a common binomial factor.When we see the same factor in both groups, we can factor it out.For quadratic expressions like this one, we can use the AC method.First, multiply the coefficients of x squared and the constant term.Then find two numbers that multiply to give thirty and add to give seventeen, the coefficient of x.We can split the middle term using these numbers.Now group the terms and factor each group.Finally, factor out the common binomial.Always verify your factoring by multiplying the factors back together.Multiply each term of the first factor by each term of the second factor.Add all terms to get back our original expression.