Hi Everyone!
On this page you will find some material about Lesson 5. Read through the material below, watch the videos, and follow up with your instructor if you have questions.
Lesson 5: Difference of Squares and ac-method
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.
- Be able to recognize a difference of squares.
- Understand the ac-method (how and why).
- Factor by factoring out the GCF if applicable and then using the ac-method.
- Communicate effectively using written and oral means.
Topic. This lesson covers
Section 4.6: Factoring Trinomials and
Section 4.7: Factoring Binomials.
WeBWorK. There are two WeBWorK assignments on today’s material:
DifferenceOfSquares
AC-Method
Lesson Notes.
Video Lesson.
Video Lesson 5 (based on Lesson 5 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
Multiply
$$(x-a)(x+a).$$
Show Answer 1
\begin{align*}&(x-a)(x+a) \\=&x^2+xa-xa-a^2\\ =&x^2-a^2\end{align*}
Warmup Question 2
Factor by grouping
$$10a^2+5a-8a-4.$$
Show Answer 2
\begin{align*}&10a^2+5a-8a-4 \\=& 5a(2a+1)-4(2a+1)\\ = &(5a-4)(2a+1)\end{align*}
Review
If you are not comfortable with the Warmup Questions, don’t give up! Click on the indicated lesson for a quick catchup. A brief review will help you boost your confidence to start the new lesson, and that’s perfectly fine.
Need a review? Check Lesson 4.
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 Difference of Squares and ac-method
Key Words. Difference of squares, ac-method, GCF (greatest common factor)
$\bigstar$ Difference of squares:
$$x^2-a^2=(x-a)(x+a).$$
$\bigstar$ The ac-method for factoring $ax^2+bx+c$
This method consists of finding two numbers whose product is $ac$ and sum is $b$. Rewrite the middle term $bx$ as the sum of two terms given by these two numbers. Then proceed factoring by grouping.
$\bigstar$ Remember: when factoring a polynomial, try to factor the GCF first.
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 factorization of the following expressions
- $x^2-9$
- $x^2y^2-25z^2$
- $27x^2-30x-8$
- $9x^2+6x-8$
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 $$9y^2-4z^2.$$
Show Answer 1
$$9y^2-4z^2=(3y-2z)(3y+2z)$$
Try Question 2
Factor $$x^2z^3-9y^2z^3.$$
Show Answer 2
\begin{align*}&x^2z^3-9y^2z^3\\=&z^3(x^2-9y^2)\\=&z^3(x-3y)(x+3y)\end{align*}
Try Question 3
Factor $$9x^2-35x-4.$$
Show Answer 3
\begin{align*}& 9x^2-35x-4\\=&9x^2-36x+x-4\\=&9x(x-4)+(x-4)\\=&(9x+1)(x-4)\end{align*}
Try Question 4
Factor $$9x^3-35x^2-4x.$$
Show Answer 4
\begin{align*}& 9x^3-35x^2-4x\\=&x(9x^2-35x-4)\\=&x(9x^2-36x+x-4)\\=&x(9x(x-4)+(x-4))\\=&x(9x+1)(x-4)\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:
DifferenceOfSquares
AC-Method
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
- How can you tell when you have a difference of squares?
- Is there a sum of squares formula?
- Demonstrate why $a^2-b^2=(a+b)(a-b)$.
- Explain with an example why the ac- method works.
$\bigstar$ Factor:
(a) $25x^2-16y^6$
(b) $6x^2z^2-31xz^2+35z^2$
Show Answer
(a) $$25x^2-16y^6 = (5x+4y^3)(5x-4y^3)$$
(b) \begin{align*}&6x^2z^2-31xz^2+35z^2\\=& z^2(6x^2-31x+35)\\=&z^2(6x^2-10x-21x+35)\\=&z^2(2x(3x-5)-7(3x-5))\\=&z^2(2x-7)(3x-5)\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.
