2. Topic VII: Polynomial Functions 7.1.1 Polynomial Operations

3. An expression which is the sum of terms of the form a x k where k is a nonnegative integer is a polynomial. Polynomials are usually written in standard form. Standard form means that the terms of the polynomial are placed in descending order, from largest degree to smallest degree. Polynomial in standard form: 2 x 3 + 5x 2 – 4 x + 7 Degree Constant term Leading coefficient The degree of a polynomial is the largest degree of its terms. When a polynomial is written in standard form, the coefficient of the first term is the leading coefficient.

4. A polynomial with only one term is called a monomial. A polynomial with two terms is called a binomial. A polynomial with three terms is called a trinomial. Identify the following polynomials: Classifying Polynomials Polynomial Degree Classified by degree Classified by number of terms 6 –2 x 3x + 1 –x 2 + 2 x – 5 4x 3 – 8x 2 x 4 – 7x 3 – 5x + 1 0 1 1 4 2 3 constant linear quartic quadratic cubic monomial binomial polynomial trinomial binomial

5. Like Terms* Definition: Monomials that are the same (the same variables to the same power) and differ only in their coefficients. Examples:

6. Adding Polynomials* To add like terms add the coefficients of both terms together Examples

7. To combine like terms To add like terms add the coefficients of both terms together Example

8. More Examples of Adding 1.( 3x 2 -4x + 8) + (2x - 7x 2 – 5) =(3x 2 – 7x 2 ) + (-4x +2x) + (8-5) = - 4x 2 – 2x + 3 2.(3n 2 – 8 +2n) + ( 5n + 13 + n 2 ) = ( 3n 2 + n 2 ) + ( 2n + 5n) + (-8 +13) =4n 2 + 7n + 5

9. Practice on Adding (2b 3 -4b +b 2 ) + ( -9b 2 + 3b 3 ) That’s correct!!!. Wrong! Try again. a. 11b 3 -10b 2 + 4b b. 5b 3 + 10b 2 -4b c. 5b 3 -8b 2 -4b

10. Subtracting Polynomials Remember subtraction means to add the opposite. To subtract polynomials you will add the opposite. To add the opposite you will change the signs of every term in the second polynomial and then add.

11. Examples of Subtracting 1. (3n 2 + 13n 3 + 5n) – (7n + 4n 3 ) = (3n 2 + 13n 3 +5n) + (-7n – 4n 3 ) = ( 13n 3 -4n 3 ) + (3n 2 ) + (5n-7n) = 9n 3 +3n 2 -2n 2. ( 6y 2 + 8y 4 -5y)-( 9y 4 -7y +2y 2 ) = (6y 2 +8y 4 -5y) + (-9y 4 +7y – 2y 2 ) = (8y 4 -9y 4 ) + (6y 2 -2y 2 ) + (-5y +7y) = -y 4 + 4y 2 + 2y)

12. Practice on Subtracting ( -4y 2 –y + 10) – ( 4y 2 + 3y + 7) a. 8y 2 + 2y + 17 That’s correct! b.-8y 2 + 2y +3 c. -8y 2 -4y+3

13. Multiplying two Binomials To multiply two binomials you use the FOIL method. F- Multiply the first term in each binomial. O- Multiply the outside terms- first one in the first binomial and last one in the second. I- Multiply the inside terms which will be the last one in the first binomial and the first in the second one. L-Multiply the last term in each binomial. Then you will combine like terms if you have any. Usually the outsides and the insides yield like terms.

14. Examples of multiplying two binomials

15. Practice on Multiplying Two Binomials 1.( x -3 ) ( x+5 )= a. x 2 + 3x + 2 c. x 2 + 8x +15 b. x 2 +2x -15

16. Multiplying a Monomial and a Trinomial ( 3x) ( 6x 2 +2x+7 )= 18x 3 +6x 2 +21x

17. Multiplying a Binomial and a Trinomial ( x -3 ) ( 2x 2 +x+5 )= 2x 3 +x 2 -6x 2 +5x-3x-15 Combine like-terms 2x 3 -5x 2 +2x -15

18. Use long division to solve the following problem

19. Algebraic long division Divide 2x³ + 3x² - x + 1 by x + 2 x + 2 is the divisor The quotient will be here. 2x³ + 3x² - x + 1 is the dividend

20. Algebraic long division First divide the first term of the dividend, 2x³, by x (the first term of the divisor). This gives 2x². This will be the first term of the quotient.

21. Algebraic long division Now multiply 2x² by x + 2 and subtract

22. Algebraic long division Bring down the next term, -x.-x.

23. Algebraic long division Now divide –x², the first term of –x² - x, by x, the first term of the divisor which gives –x.–x.

24. Algebraic long division Multiply –x by x + 2 and subtract

25. Algebraic long division Bring down the next term, 1

26. Algebraic long division Divide x, the first term of x + 1, by x,x, the first term of the divisor which gives 1

27. Algebraic long division Multiply x + 2 by 1 and subtract -x -2

28. Algebraic long division The remainder is –1. -x -2

29. Time for you to get to work!

30. Try This ONE! Check

31. Try This One!