- How do you factor the difference of two squares? How do you factor the perfect square trinomial? How do you factor the sum and difference of two cubes? Which of these three makes the most sense to you? Please give examples of each.Post a 100-200 word response
- How would you explain how to factor a trinomial with the leading coefficient greater than one to a person just learning this concept. Please provide an example.Post a 100-200 word response
Respond to the students:
- Factoring the difference between two squares:
Example:
4y^2 - 36y^6
4y^2: common factor
4y^2(1 - 9y^4): factored out
(1 - 9y^4): 1 and 9y^4 are perfect squares
1 * 1 = 1
9y^4 = 3y^2
(1 + 3y^2) and (1 - 3y^2): factors
4y^2(1 + 3y^2)(1 - 3y^2)
How do you factor the perfect square trinomial?
When factoring perfect square trinomial remember (a + b)^2 = a^2 + 2ab + b^2 and (a - b)^2 = a^2 -2ab + b^2
Example: x^2 + 12x + 36: factor, it fits the pattern because x^2 and 36 are perfect squares and 12x is twice the product of x and 6.
(a + b)^2 = a^2 + 2ab + b^2: All signs are positive
a = x, b = 6
(x + 6)^2
How to factor the difference of two perfect cubes?
To factor the difference of two perfect cubes you must remember the difference of two perfect cubes equals the difference of their cube roots multiplied by the sum of their squares and the product of their cube roots.
Example: 216-125
a^3 -b^3 = (a -b)(a^2 + ab + b^2)
a^3 = 216, b^3 = 125: the cube root of 216 is 6 and cube root of 125 is 5.
216-125 = (6 - 5)(36 + 30 + 25): substitute the values into the equation.
I would have to say that the one that makes the most sense to me would be factoring the perfect square trinomial.
4. The way to factor the difference of two perfect squares is to remember, if subtraction separates two squared terms, then the sum and the difference of the two square roots factor the binomial. An example is a2- b2=(a+b)(a-b).If we factor 9x2-16 you must first find the square root of the two terms that are perfect squared.
The square root of 9x2-16=(3x+4)(3x-4).
To factor the perfect square trinomial you can us the form a2+ab+b2. You can see that a and b contains numbers and variables. Because of this you can factor it as (a+b)2, which is a perfect square.
To factor the different of two cubes you have to remember that the difference of the two cubes equals the difference of the cube roots which would be multiplied by the sum of their squares and the product of their cube roots. An example would be a3-b3=(a-b)(a2+ab+b2). The sum of the cubes equals the sum of its roots times the squares of it roots subtracted by the product of the roots a3+b3=(a+b)(a2-ab+b2). I would say that the one that make the most sense to me is the difference of two squares.
5. Ok so I hope I am saying this the right way, please let me know if I am not. When the sum of two different numbers can be multiplied by their difference, (a+b)(a-b), then their product should be the difference of the squares. (a-b)(a+b) = b2-a2, so the terms that are alike will cancel out. Also lets take another example, 4w^2+13x+3 this would equal out to be (4x+1)(x+3). Also here is a great site that I found that has some more information on it that should be able to help.
6. Good Evening Lynn and Class,
They way I would explain how to factor a trinomial with the leading coefficient greater than one to a person just learning the concept would be to take it step by step so not to overwhelm the person which could then discourage them from wanting to continue to listen. I would say to them that you want to first take the trinomial and break it down in its standard form ax^2 + bx+ c (Note: The leading coefficient is "a" and a, b, c are real numbers) or to use a real example x^2 + 13x +36 and try to determine the integer pair with the given product and sum for the number which represents "c" in this case that would be 36. Additionally, the integer pair has to also sum up to the number which represents "b" in this case would be 13. To determine the integer pair for 36 means trying to find all its factors and in this case those pairs are (1,36), (2,18), (3,12), (4,9) and (6,6). Once you have list all of the factors for the number 36 then you want to sum up each pair and whichever pair adds up to the number 13 will be the greatest common factor with the leading coefficient greater than one. Finally, I would show this person a couple videos as they always help further explain how to solve a math problem.