Online math tutoring lessons are greatly beneficial to children who struggle in math class. With new, challenging topics to understand every year, it’s no wonder that some kids need some extra help from time to time. Solutions to certain problems in math class can be taught one way in school, while there may be additional ways to attack the problems. Often, students who use math tutoring services are introduced to new ways to understand and solve familiar math problems.
One set of math problems that can be answered in multiple ways is factoring polynomials. At first glance, this may seem like a complex mathematical concept, especially for kids who have never seen the words ‘factor’ and ‘polynomial’ before. In reality, these problems are not too complicated, and there are a couple of ways that they can be solved.
A polynomial is a mathematical expression that consists of variables, coefficients, and exponents. One example of a polynomial is 3x² + 2x – 5. In this polynomial, “x” is our variable, and the highest exponent – which is the number that indicates the degree of each term – is the 2 in 3x². This is important because the degree of the entire polynomial is determined by the highest exponent. Then, we have our coefficients, which are 3, 2, and -5.
Independent of our polynomial, we must understand the term ‘factoring’. Factoring is simply the process of finding out what to multiply together to get an expression. Our polynomial expression has factors, and by factoring a polynomial, we can identify these factors.
The ‘Bracketing the Common Factor’ Method
Now that we understand what a polynomial is, what each of its components is, and what factoring means, we can dive into the first of our methods: the ‘bracketing the common factor’ method. Not the most elegant of names, but an extremely accurate one. This method can only be used when all the terms of a polynomial share a common factor, and it is the easier of the two methods.
First, we need to identify the greatest common factor (GCF) of all of the terms in the polynomial. The GCF is the largest number that can evenly divide all of the coefficients in our polynomial.
Next, we need to factor out the GCF from each term. What this does is simplify the polynomial. Let’s use the polynomial 4x² + 8x as an example. Here, the GCF is 4x because each term has an “x” and the GCF of 4 and 8 is 4.
Finally, we must express the polynomial as the GCF multiplied by a simplified expression. Continuing with the same example, we are left with:
(4x² + 8x) / 4x = 4x (x+2)
Therefore, the factors of 4x² + 8x are 4x and (x+2).
The ‘Grouping’ Method
The second method we can use to factor polynomial expressions is called the ‘grouping’ method. This approach is useful when there is no common factor among all the terms of the expression. What we do in this case is strategically group the expressions together.
Put the terms into pairs. If you have an odd number of terms, you can group them in whatever manner you see fit. We will use a new example this time, which is 3x² + 4x – 6x – 8. We will group the terms as follows: (3x² + 4x) – (6x + 8).
Next, we must find the GCF for each group of expressions that we’ve created. In our example, the GCF of (3x² + 4x) is x, while the GCF of – (6x + 8) is -2.
Now, we need to examine the binomials that we were left with in step 2 and see if they share any common factor. If they do share a common factor, we can factor it out.
Therefore, we can say that the factors of (3x² + 4x) – (6x + 8) are (x + 1) and (3x – 2).
At Online Math Center
As demonstrated, factoring polynomials is not as tricky as it first seems. It is, however, a skill that improves only with practice. As middle school students work on different polynomials, they will become more skilled at quickly identifying common factors and utilizing the group method where needed. Getting your kids the practice they need to excel at factoring polynomials doesn’t need to be stressful, either.
At Online Math Center, we have years of experience helping middle school students grasp concepts such as factoring polynomials. In our tailored one-to-one tutoring lessons, students get the attention they deserve and the opportunity to focus on the topics they’re struggling with. We also offer prep classes for the Math SAT and math competitions, as well as classes aimed at high school students.