Wednesday, April 20, 2011

Chapter 11.3 Factoring

At the start of class, Mr. Cope did his daily routine of checking in everyone's homework. After that we began a new section called Factoring. We learned about factoring a polynomial, which is the process of rewriting a polynomial as a product of two or more factors. Mr Cope described to the class that there were 3 special cases of factoring:

1) The Greatest Common Factor (GCF)

Example: Factor 5x^3 - 15x^2

To solve, look for the greatest common monomial factor of the terms. In this case, the GCF of 5 and 15 is 5. X^2 is the highest power of x that divides each term. So 5x^2 is the greatest common monomial factor of 5x^3 and -15x^2. The last step is to apply the Distributive Property.

5x^3 -15x^2= 5x^2(x-3)

to check the answer, you can use the 'expand' command on the NSPIRE calculator by clicking MENU, 3, and then 1.

5x^2(x-3). 5x^2(x-3)= 5x^2 * x - 5x^2 * 3 = 5x^3 -15x^2

2) Difference of Perfect Squares

Example: 121 - y^2
= 11^2 - y^2

= (11+y)(11-y)

To solve a problem like this

3) Perfect Square Trinomial

Example: (8x + 2)^2 = (8x + 2)(8x + 2) =

(8x * 8x) + (8x * 2) + (2 * 2) =

64x^2 +16x +16x + 4

So if you add like terms the answer is 64x^2 +32x+4

After this, we began working on problems on the factoring worksheet he handed out to everybody. Before we began working on the problems, Mr. Cope gave the class the tip to guess and check on problems like x^2 -2x-15.

You want solutions to x^2 -2x-15=(x+____)(x+____)

Because -15 is the product of the the two missing numbers on the right side, you should think of the factors of -15. These factors are either -15 and 1, 15 and -1, 5 and -3, and -5 and 3. Keep filling in the blanks until you get the right answer.

(x-15)(x+1)=x^2 -14x-15 NO!

(x+15)(x-1)=x^2+14x-15 NO!

(x+5)(x-3)=x^2+2x-15 NO!

(x-5)(x+3) = x^2 -2x-15 YEP!

so this means that x^2 -2x-15 = (x-5)(x+3)

On the worksheet, there were many good examples for practice.

For example:

1. X^2 -81

The answer is (x-9)(x+9) because when you factor this using FOIL, you multiply x*x which = x^2. You then multiply x*9 which = 9x. Next you multiply -9x*x which =-9x. Finally you multiply -9*-9 which = 81.

You add like terms and you get x^2 -81

2. x^2 +14x +49

The answer is (x+7)^2 because when you foil this out, you multiply x*x which = x^2. You then mulitply the outsides, which is 7*x=7x. Third, you multiply insides which again is 7*x=7x. The last thing you do is multiply the last terms. For this, you get 7*7 which =49.

You add up the like terms and you get x^2+14x+49

Hopefully this helps you understand polynomial factoring a little bit more, and if it didn't, don't forget Mr. Cope is always willing to help if you go to him!

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