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# March 17, 2014

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### March 17, 2014

1. 1. Today:  Make-Up Tests?  Expanding vs. Factoring Polynomials  Polynomial Long Division  Class Work: Show all Work!!  Re-bubble answer sheets from last test
2. 2. Solving equations and factoring are the two most important concepts in algebra. There are concepts we will cover later in which the first step will be to factor a polynomial. If you can’t factor the polynomial then you won’t even be able to start the problem, let alone finish it. For example.. Class Notes: Intro. to Factoring Simplify: (x2 + 2x – 3) (x2 + 6x + 9)
3. 3. Polynomials: Expanding vs. Factoring Expanding Polynomials: is simply the multiplication (distribution) of each term by every other term. At this point, we can expand most polynomials. by which we determine what we multiplied to get the given quantity. This we do this all the time with numbers. 8 • 7 = 56 Factors Product Factoring Polynomials: factoring is the process A common method of factoring numbers is to completely factor the number into positive prime factors. If we completely factor a number into positive prime factors there will only be one way of doing it.
4. 4. Only one of the above factorizations of 12 is a complete factorization. That is the prime factorization of 12. Factoring polynomials is done in the same manner. We determine all the terms that were multiplied together to get the polynomial. We then try to factor each of the terms we found in the first step. This continues until we simply can’t factor anymore. When we can’t do any more factoring, the polynomial is completely factored. Factoring Polynomials For instance, here are just a few of the ways to factor 12.
5. 5. Here are four examples: x2 - 20x + 100 = x2 - 16 = x2 +10x + 25 = 2. If you write a polynomial as the product of two or more polynomials, you have factored a polynomial. For example: Summary: 3. A fully factored polynomial cannot be simplified further. 4. Not all polynomials can be factored!! (Using Whole Numbers) 1. Expanding & Factoring are essentially opposite operations. 5. We will start by factoring the 3 special products because of their simple, easy to memorize factor patterns. 34 – 9x(x + 2) x2 + 12x + 35 = (x + 5)(x + 7)
6. 6. Square of a sum Square of a difference Difference of Squares Special Products Find each Product (Expand) Factor:
7. 7. Dividing Polynomials Dividing a Polynomial by a binomial Please take complete, easy to read notes, you will need them.
8. 8. Dividing Polynomials Part I: Dividing by a Monomial: 1. Take each term in the numerator and divide by the denominator. The result is now a monomial divided by a monomial for each term. 18x4 -10x2 + 6x7 2x2 Part II: Dividing by a Polynomial: (Long Division) We will divide x3 + x2 - 5x -2 by x-2; x-2 x3 + x2 - 5x -2 Step 1: Write both in standard form. Then find the highest degree terms in both the divisor and the dividend. In this case, that would be x and x3 . Solve.
9. 9. Dividing Polynomials Step 3: Multiply by the Divisor, subtract: Step 2: Divide x3 by x. The result is: Place above the x2 in the dividend 3x2 - 5x-2 Step 4: Repeat the Process: 3x2 ÷ x = 3x2 -6x Step 5: Multiply by the Divisor:x- 2 x- 2 0 Step 6: Last Division x-2 x3 +x2 - 5x -2 x2 x3 - 2x2 + 3x +1 Quick Check. Multiply: (x – 2)(x2 + 3x +1)=
10. 10. Dividing Polynomials x-2 x2 - 4 = **Note: If there are missing degrees, fill in with 0𝑥 𝑑𝑒𝑔𝑟𝑒𝑒 Remainder: If there is a remainder, it is shown last as the remainder over the divisor. Solve. x-2 x2 + 0x1 - 4 Solve.
11. 11. Class Work: See Handout