The document is about factoring polynomials, specifically factoring the sum and difference of cubes. It provides the formulas for factoring the sum and difference of cubes, along with examples of factoring expressions using those formulas. It also discusses factoring out the greatest common factor from polynomials.
This will help you in factoring sum and difference of two cubes.
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This will help you in factoring sum and difference of two cubes.
For more instructional resources, CLICK me here!
https://tinyurl.com/y9muob6q
LIKE and FOLLOW me here!
https://tinyurl.com/ycjp8r7u
https://tinyurl.com/ybo27k2u
This powerpoint presentation discusses or talks about the topic or lesson Functions. It also discusses and explains the rules, steps and examples of Quadratic Functions.
This powerpoint presentation discusses or talks about the topic or lesson Functions. It also discusses and explains the rules, steps and examples of Quadratic Functions.
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2024.06.01 Introducing a competency framework for languag learning materials ...Sandy Millin
http://sandymillin.wordpress.com/iateflwebinar2024
Published classroom materials form the basis of syllabuses, drive teacher professional development, and have a potentially huge influence on learners, teachers and education systems. All teachers also create their own materials, whether a few sentences on a blackboard, a highly-structured fully-realised online course, or anything in between. Despite this, the knowledge and skills needed to create effective language learning materials are rarely part of teacher training, and are mostly learnt by trial and error.
Knowledge and skills frameworks, generally called competency frameworks, for ELT teachers, trainers and managers have existed for a few years now. However, until I created one for my MA dissertation, there wasn’t one drawing together what we need to know and do to be able to effectively produce language learning materials.
This webinar will introduce you to my framework, highlighting the key competencies I identified from my research. It will also show how anybody involved in language teaching (any language, not just English!), teacher training, managing schools or developing language learning materials can benefit from using the framework.
2024.06.01 Introducing a competency framework for languag learning materials ...
2/27/12 Special Factoring - Sum & Difference of Two Cubes
1. Bell Ringer Factor the following: x 2 – 81 x 2 +10x +25 1. 2. Students will be able to factor polynomial equations. Page 356 #3-17
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Editor's Notes
When factoring, look at what perfect squares you have and pay attention to your signs. x^2 -81 = (x-9)(x+9). When factoring this, we know that 81 is a perfect square. Since there is no middle term and 81 is negative, this can be factored into a difference of squares. x^2+10x+25 For this one, we once again have a perfect square, 25. 25 = 5^2. But, we have a middle term which is positive. This is why we can factor this quadratic into two perfect squares. For more practice, reference Sections 4.3 & 4.4 Specifically, page 253 for these special cases.
A cube is any number or variable raised to the third power. So, 2^3 = 8, making 8 a cube. The same applies for variables, such as x^3. This would also be a cube. We’ll take a look at factoring in the next couple slides. Factoring is similar to when we factor a quadratic, except this time, we factor into a binomial and a trinomial. The purpose of factoring is to solve our equations and find our zeros.
Do you notice a similarity between the two factorizations? What is similar? What is different?
Example 1 How can we check that our factors are correct? You can always double-check your work by multiplying out the polynomial (which is what we did in the last section)!
Exercise 1: Difference of cubes x^3 -125 = (x-3)(x^2 +5x+25) Exercise 2: Sum of cubes x^3 + 216 = (x+6)(x^2-6x+36)
Now, consider what you would do when we have a coefficient in front of our first term. Well…what would you do if you were told to write that coefficient in terms of a cube? Let’s take a look. Exercise 3: 8x^3 -1 We know that 8 is a cube. 8 may be written as 2^3. So, we can write our polynomial as… (2x)^3 – (1)^3 In this case, our a =2x and b=1. Now, we can use our difference of cubes formula. 8x^3-1 = (2x-1)(4x^2+2x+1) * Common mistake - Be careful and make sure that you are squaring your a correctly…so that you square the two as well to get 4x^2. Exercise 4: Sum of cubes Once again, write your polynomial in terms of cubes! 27x^3 +343 = (3x)^3 + (7)^3 So, when we use our formula, we get (3x+7)(9x^2-21x+49)
Also included in this section is factoring out a common monomial (aka the most common factor). This should be a review from prior, only you probably did not do this with as many terms, and some do require further factoring. Example: 3x^4 +9x^2 -6x Look at each term in our polynomial. First, I like to look at our numbers…3, 9, 6. What is in common between each of these? Since a three is in common, we can factor that out of each term. Now, look at our variables. We have x^4, x^2, x. Clearly, x is the common factor, so we will factor that out. Caution: When doing this, be very careful and pay attention to your powers and signs. It is really easy to drop a power or forget to factor out of each term. 3x^4+9x^2-6x = 3x(x^3+3x-2).
Exercises: Note, some require further factorization, such as factoring for difference of two squares or perfect square trinomials, like we did in the bell ringer. Exercise 5: 3x^3+9x^2-81 = 3(x^3+9x^2-27) Exercise 6: 4x^4 – 16x^3 +16x^2 Careful, this polynomial can be factored further. = 4x^2 (x^2 -4x + 4) =4x^2 (x-2)(x-2) or 4x^2(x-2)^2 Exercise 7: 2x^5-18x^3 = 2x^3 (x^2 - 9)
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