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Unit 2, Lesson 3

Unit 2, Lesson 3. Factoring. General Rules for Factoring. The first step in factoring any polynomial is to factor out the GCF (greatest common factor) The next step is to determine how many terms there are in the polynomial. The number of terms will guide which strategy you use to factor.

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Unit 2, Lesson 3

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  1. Unit 2, Lesson 3 Factoring

  2. General Rules for Factoring The first step in factoring any polynomial is to factor out the GCF (greatest common factor) The next step is to determine how many terms there are in the polynomial. The number of terms will guide which strategy you use to factor

  3. Factoring Polynomials with Two Terms- Binomials There are really only a couple options for factoring a binomial The first is if the binomial is a difference of squares  a2 - b2 note that you might have to re-write the binomial to have two “squares” For example, x2 - 25 is a diff of squares  x2 - 52 To factor it, remember that a2 - b2 = (a – b)(a + b) So, x2 - 25 = x2 - 52 = (x – 5)(x + 5)

  4. Factoring Polynomials with Two Terms- Binomials More examples: a2 – 49 = a2 – 72 (a – 7)(a + 7) y2 – 64 = (y – 8)(y + 8) c2 – 9 = (c – 3)(c + 3) 4x2 – 100 = (2x – 10)(2x + 10)

  5. Factoring Polynomials with Two Terms- Binomials The other option to factor a polynomial with two terms is if it is a difference or a sum of cubes While we won’t use this often, it does come up: a3 – b3 = (a – b)(a2 + ab + b2) a3 + b3 = (a + b)(a2 - ab + b2) These formulas are on the inside page of your textbook and you will be able to use this page on quizzes and tests from now on

  6. Factoring Polynomials with Three Terms- Trinomials This was covered in the last lesson – The easiest trinomial to factor is a PST (perfect square trinomial) so always check for this first PSTs have first and third terms that are squares and the middle term is two times the roots of the first and third terms Ex. x2 + 10x + 25 = (x + 5)2 Ex. t2 - 16t + 64 = (t - 5)2

  7. Factoring Polynomials with Three Terms- Trinomials Another way to determine if a trinomial is a PST is to take half of the middle term coefficient and square it. If your answer is the third term, it’s a PST Ex. x2 + 6x + 9  ½ of 6 is 3; 32 = 9 so it is a PST What would the third term have to be for the following to be PSTs? x2 + 18x + ? 81 x2 + 12x + ? 36 x2 + 4x + ? 4

  8. Factoring Polynomials with Three Terms- Trinomials If a trinomial is not a PST, you are left to do some “guess and check” work Remember that multiplying two binomials often times yields a trinomial, so you can guess that your answer will be two binomials (if the trinomial can be factored) Ex. x2 + 11x + 24  this is not a PST The answer will look like (x + )(x + ) To find the second terms in each binomial, you need to find two numbers that when multiplied give 24 and when added are 11 So: x2 + 11x + 24 = (x + 8)(x + 3)

  9. Factoring Polynomials with Three Terms- Trinomials A couple of things to keep in mind: • If the third term is negative, then one of the binomials uses subtraction and one uses addition Why? Ex. x2 + x – 6 = (x + 3)(x – 2) Ex. x2 - 2x – 8 = (x + 2)(x – 4) • Not all trinomials can be factored

  10. Factoring Polynomials with Four terms or More The key here is to look at the terms in groups of twos or threes. Then, use the factoring strategies you already know: 1. GCF • Two terms- diff of squares; sum/diff of cubes • Three terms- PST or “two parentheses” method

  11. Factoring Polynomials with Four terms or More Ex. 3x3 + 5x2– 6x -10 To begin, factor just the first two terms: 3x3 + 5x2  the GCF is x2 so: x2 (3x + 5) Next, factor the second two terms: – 6x -10  the GCF is -2 so: -2(3x + 5) Notice that you get (3x + 5) in both cases, and so that is one factor and the other factor is (x2 – 2) So, 3x3 + 5x2– 6x -10 = (3x + 5)(x2 – 2)

  12. Factoring Polynomials with Four terms or More Last example: 4x3 – 8x2 + 3x – 6 Take the GCF of the first two terms: 4x2(x – 2) Take the GCF of the second two terms: 3(x – 2) Use (x – 2) as one factor and (4x2 + 3) as the other factor 4x3 – 8x2 + 3x – 6 = (x – 2)(4x2 + 3)

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