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Summer 2016 - Session 2 Math 1300 FUNDAMENTALS OF MATH Section #16535 Monday - Friday, 10am-12pm Instructor: Dr. Angelynn Alvarez [email protected] Section 4.1 – Greatest Common Factors & Factoring by Grouping Section 4.1 – GCFs & Factoring by Grouping - Recall: In section 1.3, we discussed that the greatest common divisor (GCF) of two integers was the biggest number which divides both numbers. - We also said that the words “factors” and “divisors” are synonyms. - We can also do the same with polynomials: We can find the greatest common factor of two polynomials --- which is the largest degree polynomial that divides both polynomials. Finding the greatest common factor of polynomials: (1) (2) (3) Find the GCF of the coefficients (like in section 1.3). Multiply the GCF of the coefficients by the variables that appear in BOTH terms. To determine the exponents: The smallest exponent of each variable is what appears in the GCF. Example: Find the GCF of 9𝑥 !" and 18𝑥 ! . Example: Find the GCF of 11𝑥 !" and 22𝑥 !" . Example: Find the GCF of 15𝑥 ! and 15. Example: Find the GCF of 21𝑥 ! and 21. Example: Find the GCF of 7𝑥 ! 𝑦 !" and 49𝑥 !" 𝑦 ! . Example: Find the GCF of 11𝑥 ! 𝑦 ! and 55𝑥 ! 𝑦 ! . Example: Find the GCF of 2𝑥 ! 𝑦 ! and 12𝑥 !" 𝑦 ! . Note: If the coefficients have nothing in common, the GCF of the coefficients is 1. Example: Find the GCF of 20𝑥 ! 𝑦 !" and 11𝑥 ! 𝑦 ! . Example: Find the GCF of 11𝑥 ! 𝑦 !" and 9𝑥 !" 𝑦 !! . Example: Find the GCF of 64𝑥 ! 𝑦 ! 𝑧 ! and 8𝑥𝑦 ! 𝑧 ! . Example: Find the GCF of 48𝑥 ! 𝑦𝑧 ! and 12𝑥𝑦𝑧 ! . Example: Find the GCF of 96𝑥 ! + 72𝑥 ! + 12𝑥 ! . *Hint: Ignore [for now] the +/−, and find the GCF of all 3 terms. Example: Find the GCF of 28𝑥 ! − 42𝑥 ! + 7𝑥 ! . *Hint: Ignore [for now] any +/−, and find the GCF of all 3 terms. Factoring Polynomials - Given any polynomial, we want to write the polynomial as a product of the GCF of the terms and another polynomial: (polynomial)=(GCF)×(another polynomial) - This process is called factoring. To factor a polynomial: (1) Ignore all addition and subtraction signs [for now] and find the GCF of all of the terms. (2) Multiply the GCF in step (1) by a polynomial that you need to get to back the original polynomial that you started with. Example: Factor the following polynomials completely. 12𝑥 ! + 12 9𝑥 ! + 9 12𝑥 ! 𝑦 − 36𝑥 ! 𝑦 ! 15𝑥𝑦 ! − 20𝑥 ! 𝑦 ! 63𝑥 ! 𝑦 − 56𝑥 ! 𝑦 ! Factoring by Grouping - If a polynomial has 4 or more terms, it helps to group terms together and factor out the common terms in each group. - In this class, we will deal with terms with just 4 terms. - We call this process “Factoring by Grouping”. How to factor by grouping: (1) (2) (3) Group the first 2 terms together and the last 2 terms together. Then factor each group. *Don’t forget the any “−“! Identify the term that is common to both groups. Multiply the common term in step (2) by the sum of the uncommon terms. Example: Factor the following polynomials completely. 5 𝑥 − 𝑦 − 𝑡(𝑥 − 𝑦) 9 𝑥 − 𝑦 − 𝑡(𝑥 − 𝑦) 𝑥 𝑥 + 11 − (𝑥 + 11) 𝑥 𝑥 + 9 − (𝑥 + 9) 𝑐𝑦 + 𝑏𝑦 + 𝑐𝑥 + 𝑏𝑥 𝑚𝑏 + 𝑦𝑏 + 𝑚𝑥 + 𝑦𝑥 𝑦𝑚 + 𝑐𝑚 + 𝑦𝑎 + 𝑐𝑎 𝑥 ! − 3𝑥 ! + 6𝑥 − 18 𝑥 ! − 7𝑥 ! + 6𝑥 − 42 𝑥 ! − 5𝑥 ! + 8𝑥 − 40 *Tip for the following examples: Factoring out a “−“ creates a “+”. That is: −𝑥𝑦 − 𝑥𝑧 = −𝑥(𝑦 + 𝑧) Example: Factor the following polynomials completely. −3𝑥 + 3𝑦 − 𝑥𝑦 − 𝑦 ! 5𝑥 + 5𝑦 − 𝑥𝑦 − 𝑦 ! 7𝑥 + 7𝑦 − 𝑥𝑦 − 𝑦 !