1. E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals B.1 E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals Polynomials KFUPM - Prep Year Math Program (c) 2009 All Right Reserved Definitions Polynomial Evaluation Polynomial Equations Algebra of polynomials Factoring Polynomials

2. E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals Definitions: An expression containing a real number a multiplied by one or more variable raised to some non-negative powers is called a term with coefficient a. For example Terms which contain the same variable raised to the same powers are called like terms. KFUPM - Prep Year Math Program (c) 2009 All Right Reserved

3. E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals Definitions: The degree of a term consisting of a constant a and no variables is zero if a ≠ 0, and undefined if a =0. The degree of a term which is a product of a constant a and one or more variables is the sum of the powers of the variables. For example The degree of –2 is zero The degree of 0 is undefined The degree of is 6 The degree of is 4. KFUPM - Prep Year Math Program (c) 2009 All Right Reserved

4. E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals Note: KFUPM - Prep Year Math Program (c) 2009 All Right Reserved Two like terms are added using the distribute law, multiplied and divided using the laws of exponents. For example Example 1

5. E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals Definition An algebraic expression consisting of a finite sum of terms in the same variable(s) is called a polynomial in those variables. The greatest degree of any term in a polynomial is called the degree of the polynomial. For example, A polynomial consisting of one, two, or three terms is called a monomial, binomial or trinomial, respectively. KFUPM - Prep Year Math Program (c) 2009 All Right Reserved

6. E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals Polynomials in One Variable KFUPM - Prep Year Math Program (c) 2009 All Right Reserved A polynomial p(x), of degree n in x consist, in general, n+1 terms and can be written

7. E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals KFUPM - Prep Year Math Program (c) 2009 All Right Reserved Example 1

8. E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals Polynomial Evaluation KFUPM - Prep Year Math Program (c) 2009 All Right Reserved

9. E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals Algebra of Polynomials KFUPM - Prep Year Math Program (c) 2009 All Right Reserved Example 2

10. E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals KFUPM - Prep Year Math Program (c) 2009 All Right Reserved Example 2

11. E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals Scalar Multiplication KFUPM - Prep Year Math Program (c) 2009 All Right Reserved

12. E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals KFUPM - Prep Year Math Program (c) 2009 All Right Reserved Example 3 Solution

13. E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals Multiplication of Polynomials KFUPM - Prep Year Math Program (c) 2009 All Right Reserved Example 4

14. E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals KFUPM - Prep Year Math Program (c) 2009 All Right Reserved Solution

15. E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals Special Products KFUPM - Prep Year Math Program (c) 2009 All Right Reserved

16. E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals KFUPM - Prep Year Math Program (c) 2009 All Right Reserved Special Products

17. E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals KFUPM - Prep Year Math Program (c) 2009 All Right Reserved Example 4

18. E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals Factoring of Polynomials KFUPM - Prep Year Math Program (c) 2009 All Right Reserved Factoring polynomials is the reverse operation to multiplication. It consists of writing the given polynomial as a product of polynomials (generally of lower degrees). When factoring a polynomial, we must first factor out the greatest common factor ( GCF) of the terms of the polynomial. This operation uses the distributive law in reverse manner.

19. E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals KFUPM - Prep Year Math Program (c) 2009 All Right Reserved Example 5 Factor out the greatest common factor of the terms of the following polynomial expressions.

20. E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals KFUPM - Prep Year Math Program (c) 2009 All Right Reserved Example 6 Factor using the special products (i) –( iv) in reverse manner (after you factor out the GCF, if any)

21. E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals KFUPM - Prep Year Math Program (c) 2009 All Right Reserved

22. E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals Factoring by Grouping KFUPM - Prep Year Math Program (c) 2009 All Right Reserved When factoring a polynomial expression of four or more terms, it may happen that grouping the terms into groups and factoring each group such that a common factor between the groups occur. Then by taking out the common factor one would get a factorization of the whole expression

23. E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals KFUPM - Prep Year Math Program (c) 2009 All Right Reserved Example 7 Factor by grouping

24. E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals KFUPM - Prep Year Math Program (c) 2009 All Right Reserved Example 8 Facto and write each of the following expressions without negative exponents

25. E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals KFUPM - Prep Year Math Program (c) 2009 All Right Reserved Solution

26. E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals Factoring of a General Trinomial KFUPM - Prep Year Math Program (c) 2009 All Right Reserved

27. E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals KFUPM - Prep Year Math Program (c) 2009 All Right Reserved Example 8 Factor each of the following trinomials: