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## Polynomials and Polynomial Functions

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**Chapter 5**Polynomials and Polynomial Functions**Chapter Sections**5.1 – Addition and Subtraction of Polynomials 5.2 – Multiplication of Polynomials 5.3 – Division of Polynomials and Synthetic Division 5.4 – Factoring a Monomial from a Polynomial and Factoring by Grouping 5.5 – Factoring Trinomials 5.6 – Special Factoring Formulas 5.7-A General Review of Factoring 5.8- Polynomial Equations**Factoring a Monomial from a Polynomial and Factoring by**Grouping § 5.4**Factors**A prime number is an integer greater than 1 that has exactly two factors, 1 and itself. A compositenumber is a positive integer that is not prime. Primefactorization is used to write a number as a product of its primes. 24 = 2 · 2 · 2 · 3**If a · b = c, then a and b are of c.**factors a·b Factors To factor an expression means to write the expression as a product of its factors. Recall that the greatest common factor (GCF)of two or more numbers is the greatest number that will divide (without remainder) into all the numbers. Example: The GCF of 27 and 45 is 9.**Determining the GCF**• Write each number as a product of prime factors. • Determine the prime factors common to all the numbers. • Multiply the common factors found in step. The product of these factors is the GCF. Example: Determine the GCF of 24 and 30. 24 = 2 · 2 · 2 · 3 30 = 2 · 3 · 5 A factor of 2 and a factor of 3 are common to both, therefore 2 · 3 = 6 is the GCF.**Determining the GCF**To determine the GCF of two or more terms, take each factor the largest number of times it appears in all of the terms. Example: a.) . Note that y4 is the highest power of y common to all four terms. The GCF is, therefore, y4.**Factoring a Monomial from a Polynomial**• Determine the GCF of all the terms in the polynomial. • Write each term as the product of the GCF and another factor. • Use the distributive property to factor out the GCF. Example: 15x4 – 5x3+25x2 (GCF is 5x2)**Factor a Common Binomial Factor**Sometimes factoring involves factoring a binomial as the greatest common factor. Example**Factoring by Grouping**The process of factoring a polynomial containing four or more terms by removing common factors from groups of terms is called factoring by grouping. Example: Factor x2 + 7x + 3x + 21. x(x + 7) + 3(x + 7) = (x + 7) (x + 3) Use the FOIL method to check your answer.**Factor by Grouping Method**• Determine if all four terms have a common factor. If so, factor out the greatest common factor from each term. • Arrange the four terms into two groups of two terms each. Each group of two terms must have a GCF. • Factor the GCF from each group of two terms • If the two terms formed in step 3 have a GCF, factor it out.**Factor by Grouping Method**Example: Factor x3 -5x2 + 2x - 10. There are no factors common to all four terms. However, x2 is common to the first two terms and 2 is common to the last two terms. Factor x2 from the first two terms and factor 2 from the last two terms.**Factoring by Grouping**Example: a.) Factor by grouping: x3 + 2x + 5x2– 10 There are no factors common to all four terms. Factor x from the first two terms and -5 from the last two terms.