1 Summer MA 15200 Lesson 6 Section P.5 *There are several good study tips in this lesson on the textbook pages. I Factoring Polynomials If two binomials are multiplied together, each polynomial is a factor of the product. The first factoring method to always try is called factoring out the greatest common factor. If a polynomial cannot be factored using integer coefficients, it is a prime polynomial. Ex 1: Factor each polynomial by factoring out the GCF. Some 4-term polynomials can be factored using what is called the grouping method. 1. Group the first two terms together and factor out the GCF. 2. Group the last two terms together and factor out the GCF. 3. If there is a binomial that is a GCF of the result, that GCF is factored out and the result is a product of two binomial factors. Ex 2: Factor each polynomial by the grouping method. 2nd factoring method 1st factoring method2 A polynomial of the form is called a difference of squares. A difference of squares factors as follows: Ex 3: Factor each polynomial completely. The following are called perfect square trinomials because each equals a binomial squared when factored (as shown in these formulas). Ex 4: Factor each. Perfect square Perfect square Double product of sq. roots of 1st term and last term 3rd factoring method 4th factoring method3 There are two ways commonly used to factor trinomials. Both require these first steps. 1. Write the trinomial in descending order. 2. Factor out any GCF, possibly including a (-) so that the leading term has a positive coefficient. Some students find success with 'trial and error', simply trying different combinations until one is found such that when FOILed, the proper trinomial results. To factor a trinomial of the form using the ‘trial-and-error’ process. Follow these steps: 1. Make your first terms have a product of (x and x) 2. Make your last terms have a product of c. 3. Find the sum of the inner and outer terms and check if it equals bx. If not, go back to steps 1 and 2 and try a different combination, until step 3 checks. Other students like to use grouping to factor a trinomial. Sometimes I call this the product/sum method. Here are the steps for grouping or the product/sum method. If a trinomial is of the form , find a pair of numbers r and s; such that the product (r)(s) equals c and the sum r + s equals b. Then the factors are . Ex 5: Factor each trinomial. Factoring trinomials is the last factoring method.4 To factor a trinomial of the form using the ‘trial-and-error’ process. Follow these steps: 4. Make your first terms have a product of . 5. Make your last terms have a product of c. 6. Find the sum of the inner and outer terms and check if it equals bx. If not, go back to steps 1 and 2 and try a different combination, until step 3 checks. Other students like to use grouping to factor a trinomial. Here are the steps for grouping. I sometimes call this the product/sum method. Follow these steps if the leading coefficient is not a 1: 1. If the trinomial is in the form , find a pair of numbers whose product is ac and whose sum is b. Call these numbers r and s. 2. Write the polynomial of the form . Use the ‘grouping’ method to factor. a) Group the first two terms together and factor out a GCF. b) Group the last two terms together and factor out a GCF. c) Now factor out the GCF of the polynomial. Use whichever method is best for you! The textbook exclusively uses the trial-and-error method. Most students have greater success if they sometimes use the product/sum method. Ex 6: Factor each completely.5 When given a polynomial to factor, this is the order you should follow. 1. Always look for a GCF first. 2. If the polynomial has two terms (binomial), see if it is a difference of squares. 3. If the polynomial has 4 terms, try the grouping method. 4. If the polynomial has 3 terms (trinomial), see if it is a perfect square trinomial. 5. If a trinomial is not a perfect square, try ‘trial-and-error’ or the ‘grouping’ method. Ex 7: Factor each polynomial completely, if
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