Hi Everyone!
On this page you will find some material about Lesson 4. Read through the material below, watch the videos, and follow up with your instructor if you have questions.
Lesson 4: GCF Factoring and
Factoring by Grouping
Table of Contents
Resources
In this section you will find some important information about the specific resources related to this lesson:
- the learning outcomes,
- the section in the textbook,
- the WeBWorK homework sets,
- a link to the pdf of the lesson notes,
- a link to a video lesson.
Learning Outcomes.
- Know what a GCF is.
- Be able to factor out a GCF.
- Be able to factor by grouping
- Know the limits of factoring by grouping
- Communicate effectively using written and oral means.
Topic. This lesson covers
Section 4.5: Greatest Common Factor and Factoring by Grouping.
WeBWorK. There is one WeBWorK assignment on today’s material:
GCF-Grouping
Lesson Notes.
Video Lesson.
Video Lesson 4 (based on Lesson 4 Notes)
Warmup Questions
These are questions on fundamental concepts that you need to know before you can embark on this lesson. Don’t skip them! Take your time to do them, and check your answer by clicking on the “Show Answer” tab.
Warmup Question 1
In the product $2\cdot 5 =10$, the numbers $2$ and $5$ are factors of $10$. We also have $1\cdot 10=10$, so $1$ and $10$ are also factors of $10$. Can you find all positive factors of 20?
Show Answer 1
$$1,2, 4, 5, 10, 20$$
Warmup Question 2
(a) Find all factors of $30$.
(b) Find all factors of $50$.
(c) Find all common factors of $30$ and $50$.
(d) Find the GCF (greatest common factor) of $30$ and $50$.
Show Answer 2
(a) $\pm 1, \pm 2, \pm 3, \pm 5, \pm 6,\pm 10, \pm 15,\pm 30$
(b) $\pm 1,\pm 2, \pm 5, \pm 10, \pm 25,\pm 50$
(c) $\pm 1,\pm 2, \pm 5, \pm 10$
(d) The GCF of $30$ and $50$ is $10$.
Warmup Question 3
Distribute
$$3x^2y^4(5x^7-2xy^2+4).$$
Show Answer 3
\begin{align*}& 3x^2y^4(5x^7-2xy^2+4)\\=&15x^9y^4-6x^3y^6+12x^2y^4\end{align*}
Quick Intro
This is like a mini-lesson with an overview of the main objects of study. It will often contain a list of key words, definitions and properties – all that is new in this lesson. We will use this opportunity to make connections with other concepts. It can be also used as a review of the lesson.
A Quick Intro to the GCF Factoring and Factoring by Grouping
Key Words. Terms, factor, GCF (greatest common factor), factor by grouping
$\bigstar$ The terms of the polynomial $2x^3-4x^2+6x$ are $2x^3$, $-4x^2$ and $6x$.
The GCF (greatest common factor) is the greatest factor of all terms.
In the case of $2x^3-4x^2+6x$, the GCF is $2x$. By factoring $2x$ out, we obtain
$$2x(x^2-2x+3).$$
$\bullet$ Another example:
$$2(x-9)-x(x-9).$$
Here the terms are $2(x-9)$ and $-x(x-9)$. The GCF is the binomial $x-9$. By factoring $x-9$ out, we obtain
$$(2-x)(x-9).$$
$\bullet$ Factoring by Grouping
This method applies to four-term polynomials. First, factor the GCF out of the four terms, if any. Then factor the GCF out of the first two terms. Factor the GCF out of the last two terms. If the two remaining factors share a common binomial factor, factor it out.
\begin{align*}&2acx^2+2adx+2bcx+2bd\\=&\underbrace{2}_{\stackrel{GCF \;of\;2acx^2,}{ 2adx,\; 2bcx \; and \;2bd}}(acx^2+adx+bcx+bd)\\=&2(\underbrace{ax}_{\stackrel{GCF \;of \;acx^2\;}{ and\; adx} }(cx+d)+\underbrace{b}_{\stackrel{GCF\; of \; bcx \;}{ and \; bd}}(cx+d))\\=&2(ax+b)\underbrace{(cx+d)}_{\stackrel{GCF \;of \; ax(cx+d) \;}{ and \; b(cx+d)}}\end{align*}
Video Lesson
Many times the mini-lesson will not be enough for you to start working on the problems. You need to see someone explaining the material to you. In the video you will find a variety of examples, solved step-by-step – starting from a simple one to a more complex one. Feel free to play them as many times as you need. Pause, rewind, replay, stop… follow your pace!
Video Lesson
A description of the video
In the video you will see the following
- $3x^2y^4(1-2xy^2+3x^2y^3)$
- the GCF of $2\cdot 3^2\cdot 5^2$ and $2^2\cdot 3^3\cdot 5$
- the GCF of $x^2$, $x^3$ and $x^4$
- the GCF of $3x^2y^4$, $6x^3y^6$ and $9x^4y^7$
- the GCF of $10x^2y^3$ and $15x^3y$
- factorization of $3x^2y^4-6x^3y^6+9x^4y^7$
- factorization of $x^2+3x+2x+6$
- factorization of $12x^2+10x-18x-15$
Try Questions
Now that you have read the material and watched the video, it is your turn to put in practice what you have learned. We encourage you to try the Try Questions on your own. When you are done, click on the “Show answer” tab to see if you got the correct answer.
Try Question 1
Factor out the GCF $$10x^2y^3-15x^3y.$$
Show Answer 1
The GCF of $$10x^2y^3-15x^3y$$ is $5x^2y$. So
\begin{align*}&10x^2y^3-15x^3y\\=&5x^2y(2y^2-3x).\end{align*}
Try Question 2
Factor out the GCF $$5x^6y^9-10x^7y^6+5x^3y^5.$$
Show Answer 2
The GCF of $$5x^6y^9-10x^7y^6+5x^3y^5$$ is $5x^3y^5$. So
\begin{align*} &5x^6y^9-10x^7y^6+5x^3y^5\\=&5x^3y^5(x^3y^4-2x^4y+1).\end{align*}
Try Question 3
Factor by grouping $$27x^2+18x-6x-4.$$
Show Answer 3
\begin{align*}&27x^2+18x-6x-4\\=&27x^2-6x+18x-4\\=&3x(9x-2)+2(9x-2)\\=&(3x+2)(9x-2)\end{align*}
WeBWorK
You should now be ready to start working on the WeBWorK problems. Doing the homework is an essential part of learning. It will help you practice the lesson and reinforce your knowledge.
WeBWorK
It is time to do the homework on WeBWork:
GCF-Grouping
When you are done, come back to this page for the Exit Questions.
Exit Questions
After doing the WeBWorK problems, come back to this page. The Exit Questions include vocabulary checking and conceptual questions. Knowing the vocabulary accurately is important for us to communicate. You will also find one last problem. All these questions will give you an idea as to whether or not you have mastered the material. Remember: the “Show Answer” tab is there for you to check your work!
Exit Questions
- What is a GCF? Give an example.
- How do you factor out a GCF?
- When should you look to factor by grouping? Is it always possible?
$\bigstar$
(a) Factor the GCF out $$100x^3y^2-6xy.$$
(b) Factor by grouping $$10x^2+5x-4x-2.$$
Show Answer
(a) The GCF is $2xy$.
\begin{align*} &100x^3y^2-6xy\\=&2xy(50x^2y-3)\end{align*}
(b)
\begin{align*}&10x^2+5x-4x-2\\=&5x(2x+1)-2(2x+1)\\=&(5x-2)(2x+1)\end{align*}
Need more help?
Don’t wait too long to do the following.
- Watch the additional video resources.
- Talk to your instructor.
- Form a study group.
- Visit a tutor. For more information, check the tutoring page.
