Factoring Polynomials Explained with Animations

Welcome to understanding polynomials! Today we'll explore terms, coefficients, and common factors.A polynomial is an expression that contains variables and coefficients, using only addition, subtraction, and multiplication operations.Let's look at this example: three x squared plus six x plus nine.Every polynomial is made up of terms. Let's break down each term in our example.The first term is three x squared. The coefficient is three, and the variable part is x squared.The second term is six x. Here, six is the coefficient, and x is the variable.The third term is simply nine. This is called a constant term because it has no variable.Now, let's learn how to find the Greatest Common Factor, or GCF, of these terms.Let's list out the factors of each term.Looking at all these factors, we can see that three is the greatest number that divides all terms.Let's factor out three from each term.First, we write three times each term divided by three.Then we can factor out the three, giving us three times the quantity x squared plus two x plus three.This is our factored expression. We've successfully factored out the greatest common factor of three.To factor x squared plus five x plus six using the AC method, let's break it down step by step.First, we identify the coefficients: a is 1, b is 5, and c is 6.Next, we multiply a and c, which gives us six.We need to find two numbers that multiply to give six and add to give five.Two and three work perfectly - they multiply to give six and add to give five.Now we split the middle term, five x, into two x plus three x.Group the terms with parentheses.Factor out the greatest common factor from each group.This simplifies to x plus two times x plus three.Let's verify our answer using the FOIL method.Adding these terms together confirms our original expression: x squared plus five x plus six.The difference of squares is a special pattern where we have two squared terms being subtracted.For example, x squared minus sixteen can be factored as x plus four times x minus four.We can identify this pattern by looking for two squared terms with a subtraction sign between them.Let's verify this factorization by expanding it back using FOIL method.Another important pattern is the perfect square trinomial.For example, x squared plus six x plus nine equals x plus three squared.In a perfect square trinomial, the middle term is twice the product of the roots, and the last term is the square of the second root.Here are some important tips for recognizing these patterns quickly.Let's look at some common mistakes to avoid when working with these patterns.Here are some practice problems for you to try. Pause the video and see if you can identify the patterns.Let's review the key points about special factoring patterns.Thanks for learning about special factoring patterns with Spark.E!