Peter Michalakes
Group Factoring (and any other variation of which) is essentially taking a problem and expanding it, eventually bringing the equation to this format: ( ) ( ) in which each set of parentheses is multiplied by each other. Addition, subtraction and division cannot be present outside the parentheses when a problem is completely factored.
In this post, I will explain how to factor by grouping and how to factor quadratic equations.
Factoring Equations
When factoring, it is good to reference equations which illustrate what combination of variables equal what combination of parentheses.
Difference and Trinomial of Squares:
Sum and Difference of Cubes:
These equations are extremely helpful when factoring. For example, when you are asked to factor a + 2ab + b, it’s already done for you in the Trinomial of Squares equation, as a + 2ab + b = (a+b)^2.
Group Factoring
Factoring is a puzzle. And like any puzzle, there is no strict guideline to solving it; instead, the solver must rely on less-common mathematical methods to factor an equation. Again, referencing the previous equations are extremely helpful.
An Example:
When asked to factor x^2 + 8x +12, the first step would be to determine what factors each term has.
Listed in the picture above are the factors of 12:
6, 2;
3,4;
1, 12.
With this information, the next step is to determine the value of two of the integers in the ( ) ( ) guideline.
Bear in mind that the factors must add up to 8 from “8x,” so therefore, the factors 6 and 2 take their place in the parentheses as none of the others add up to 8. Therefore, we are left with ( + 6) ( + 2).
Now we must take the variable into account. Because x is squared, we automatically need two x’s, and since 6x + 2x = 8x, (foiled) then x is the missing term.
And so x^2 + 8x +12 factors out completely to (x+6)(x+2).
Recap
Factoring by grouping can be tedious at times. But it can become less tedious if the Trinomial of Equations, Differences of Equations, Difference of Squares and Sum and Difference of Cubes equations are used.
The general steps for factoring by grouping is:
-identify common factors.
-identify correlating equation (if possible).
-factor.
Factoring Quadratics
It can generally be said that factoring quadratics takes less time than that of equations, even more so if (I know, I need to stop bringing up the guideline equations) the Trinomial of Squares equation is used as a reference.
An Example:
Quadrics can be solved similarly to those equations involving grouping, with a similar mindset of breaking down the puzzle of an equation on the paper in front of you.
A method we had discussed in class involved making a t-chart representing the Greatest-Common-Factors between the two groups of a quadratic. The results of which are described in the picture above.
Another Example:
In class on Friday we factored several quadratic equations, many of which are listed above.
For example, in x^2 - 12x + 11, the Trinomial of Squares reference takes its form. The method of which is described earlier in this post.
And so.
Thursday and Friday’s math classes focused solely on Group Factoring and Quadratic Factoring, both of which, while requiring more thinking than some of the other units we have studied, are very helpful in expanding our mathematical education.
Note: I have no idea why the pictures made this posts' formatting messed up, but I can't seem to fix it
ReplyDeleteThis is really well written and thorough. You really made sure to cover everything we talked about. And I really appreciated the pictures of the formulas. Good job!
ReplyDeleteI found it to be very useful that you inserted pictures of the formulas so they were easy to reference back to. I also found your explanations to be clear and to the point. Thanks!
ReplyDeleteThis was really clear. I found the logical sequential steps very helpful. I also liked how you kept referring to the various helpful factoring equations that should be considered when factoring, for example, the trinomial squares. So, all in all, a really complete overview.
ReplyDeleteThis was really clear. I found the logical sequential steps very helpful. I also liked how you kept referring to the various helpful factoring equations that should be considered when factoring, for example, the trinomial squares. So, all in all, a really complete overview.
ReplyDeleteYour explanations and use of pictures really helped me understand anything i was previously confused about, Great job.
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ReplyDeletePeter, Your post starts off well and includes some good information on factoring quadratics. But, when you move to factoring by grouping, both the descriptions and organization of material are a bit confusing. For instance, your "general steps" for factoring by grouping don't seem accurate to me and they don't give the reader enough detail to follow the example in the picture. Particularity when looking back at a post to study, these kids of details are helpful. Finally, your post does not include an element from Category 3 of the rubric: a YouTube or Khan Academy video, an external website, or a reference to where the material can be found in the book would have enhanced your post.
ReplyDelete