When it comes to match, another crucial skill to have would be the ability to quickly factor and expand various polynomials. In this post, it is my goal to go over factoring and expanding of everything from the basic gcf stuff to quadratic equations.
Let’s begin with factoring and expanding these kinds of expressions:
To expand an expression, simply multiply the outer term with everything inside the brackets. For factoring however, we must find the gcf or greatest common factor of the coefficients and constants. Also, if there are variables with with an exponent larger than 1, we will take the variable with the smallest exponent on it. For example, if we were to factor x^4 – x^3 + x^2, we would take x^2 because that is the most we can factor out of every term – giving us: x^2(x^2-x+1). Now, let’s apply these ideas to our problems:
Now that we have factored and expanded some of the basic expressions, let’s move on to examples where we are multiplying a binomial (something like a + b) with a binomial. Similarly, we will factor the same kinds of multi-step expressions. Here are the questions:
Before we move on with the questions, it’s important to review what FOIL is. Well, it is an acronym for the steps we take to expand out a binomial x binomial expression. In general, when it comes to multiplying terms together, make sure that every term from one bracket multiplies with every term from the other. Now, with this in mind, let’s look at our problems:
Now, using FOIL and other techniques mentioned above, let’s factor and expand these expressions!
Basically, with factoring these more complex expressions, just go back and check for any common factors. For example with question 2, we realized that (-v + r) was a common factor and then we further simplified our expression. Lastly, we will go over the more challenging examples of factoring/expanding, such as a difference of squares and factoring quadratic equations. Here are our base problems:
To be able to factor these expressions, we must first go over some of the basics for each type of expression. For a difference of squares expression, such as a^2 – b^2, convert it into (a + b)(a – b). For quadratic expressions such as ax^2 + bx + c, where a = 1, find two values that multiply to c and add to b. Now, put that into: (x + r)(x + s), where r and s are the values that satisfy those restrictions. Finally, for quadratic expressions where a ≠ 1, find two values that multiply to ac and add to b. Then, convert your + bx term into two terms that have coefficients resembling the two values you found. (Sometimes the order of these 2 terms has to changed…) Then, you will factor the first and second terms, and then factor the third and fourth terms. You should be able to see 1 common term and then you can factor that out as well. It will make more sense with the solutions:
To be honest, factoring can be only difficult when you forget to go back and look for factors even after you factored something. Hence, let’s conclude with some problems that require us to go back and look for even more factors:
Now, using what we’ve learned, let’s solve these problems!
Overall, we learned a lot about factoring and expanding. More importantly, we looked at some of the common expressions and how to properly factor them. In conclusion, look for gcf’s and these kinds of special expressions that can factor into something else. Keep on doing this until you can’t possibly factor it further!