Unit 1: Polynomials. Expressions: - mathematical sentences with no equal sign. Example: 3x + 2

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1 Pure Math 0 Notes Unit : Polynomials Unit : Polynomials -: Reviewing Polynomials Epressions: - mathematical sentences with no equal sign. Eample: Equations: - mathematical sentences that are equated with an equal sign. Eample: 5 Terms: - are separated by an addition or subtraction sign. - each term begins with the sign preceding the variable or coefficient. Numerical Coefficient Monomial: - one term epression. Binomial: - two terms epression. Eample: 5 Eample: 5 5 Eponent Variable Trinomial: - three terms epression. Eample: 5 Polynomial: - many terms (more than one) epression. All Polynomials must have whole numbers as eponents!! Eample: 9 is NOT a polynomial. Degree: - the term of a polynomial that contains the largest sum of eponents Eample: 9 y 5 y Degree (5 ) Eample : Fill in the table below. Polynomial Number of Terms Classification Degree Classified by Degree 9 monomial 0 constant monomial linear 9 binomial linear trinomial quadratic 9 polynomial cubic 9 trinomial quartic Like Terms: - terms that have the same variables and eponents. Eamples: y and 5 y are like terms y and 5y are NOT like terms Copyrighted by Gabriel Tang, B.Ed., B.Sc. Page.

2 Unit : Polynomials Pure Math 0 Notes To Add and Subtract Polynomials: Combine like terms by adding or subtracting their numerical coefficients. Eample : Simplify the followings. a. 5 b. (9 y y ) ( y 0 y ) 5 9 y y y 0 y 9 y y c. (9 y y ) ( y 0 y ) 9 y y y 0 y (drop brackets and switch signs in the bracket that had sign in front of it) 9 y y d. Subtract 9 5 This is the same as (9 ) (5 ) 9 5 To Multiply and Divide Monomials: Multiply or Divide (Reduce) Numerical Coefficients. Add or Subtract eponents of the same variable according to basic eponential laws. Eample : Simplify the followings. a. ( y ) ( y y z ) b. 5 yz ()() ( )( ) (y )(y 5 y z ) 5 y z 5 5a b c. 5 5a b 5 a b 5 5 a b 5 y y z 0 ( z 0 ) a b or b a y Page. Copyrighted by Gabriel Tang, B.Ed., B.Sc.

3 Pure Math 0 Notes Unit : Polynomials (AP) Eample : Find the area of the following ring. General Formula for Area of a Circle A πr Inner Circle Radius Outer Circle Radius ( ) Inner Circle Area: A π () A π ( ) A π Outer Circle Area: A π () Shaded Area π π Shaded Area π A π ( ) A π - Homework Assignment Regular: pg. 0-0 # to 5, 55, 5 AP: pg. 0-0 # to 5, 5-5 Copyrighted by Gabriel Tang, B.Ed., B.Sc. Page.

4 Unit : Polynomials Pure Math 0 Notes -: Multiplying Polynomials To Multiply Monomials with Polynomials Eample : Simplify the followings. a. ( ) b. ( ) ( ) ( ) c. (5 ) ( ) d. (a a ) (a ) (only multiply (5 ) ( ) the brackets (a a ) (a ) right after the 5 monomial) a a a 5a a To Multiply Polynomials with Polynomials Eample : Simplify the followings. a. ( ) ( ) b. ( ) ( 5 ) ( ) ( ) ( ) ( 5 ) c. ( ) ( ) ( ) ( ) d. ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( 5 ) Page. Copyrighted by Gabriel Tang, B.Ed., B.Sc.

5 Pure Math 0 Notes Unit : Polynomials Eample : Find the shaded area of each of the followings. a. 5 b. Shaded Area Big Rectangle Small Square (5 ) ( ) ( ) ( ) 5 (0 5 ) ( ) (0 ) ( ) 0 Shaded Area ( ) ( ) Total Area Top Rectangle Bottom Rectangle ( ) ( ) ( ) ( 5) ( ) ( 0 5) ( ) ( 9 5) 9 5 Total Area Homework Assignment Regular: pg # to (odd),, AP: pg # to (even), 5,,, 9 Copyrighted by Gabriel Tang, B.Ed., B.Sc. Page 5.

6 Unit : Polynomials Pure Math 0 Notes -: Special Products ( y) ( y) ( y) ( y) is NOT y ( y) ( y) ( y) ( y) Eample : Simplify the followings. a. ( ) b. ( ) ( ) ( ) ( ) ( ) ( ) 9 ( ) ( ) 9 ( ) ( ) c. ( ) ( ) Eample : Find the volume of the bo below. ( ) ( ) ( ) ( ) (9 ) ( ) (9 ) ( ) 9 5 Volume Length Width Height - Homework Assignment Regular: pg.- # to (odd), 9, 5, 5 (a, c, e, g), 55 (a, c, e, g), 5 AP: pg.- # to (even), 9 to 5, 5 (b, d, f, h), 55 (b, d, f, h), 5, 5 V ( ) ( ) V (9 ) ( ) V (9 ) ( ) V 9 Volume Page. Copyrighted by Gabriel Tang, B.Ed., B.Sc.

7 Pure Math 0 Notes Unit : Polynomials -: Common Factors Common Factors can consist of two parts: a. Numerical GCF: - Greatest Common Factor of all numerical coefficients and constant. b. Variable GCF: - the lowest eponent of a particular variable. After obtaining the GCF, use it to divide each term of the polynomial for the remaining factor. Eample : Factor the followings a. b. a b ab ab GCF ( ) ab (a b ) GCF ab Factor by Grouping (Common Brackets as GCF) a (c d) b (c d) (c d) (a b) Common Brackets Take common bracket out as GCF Eample : Factor. a. ( ) ( ) b. ab ac b bc ( ) ( ) (ab ac) (b bc) a (b c) b (b c) c. y 9 y (b c) (a b) ( y ) (9 y ) ( y ) (9 y ) Brackets are NOT the same! We might have to first rearrange terms. Try again after rearranging terms! 9 y y ( 9) (y y ) ( ) y ( ) Switch Sign in Second Bracket! We have put a minus sign in front of a new bracket! ( ) ( y ) Copyrighted by Gabriel Tang, B.Ed., B.Sc. Page.

8 Unit : Polynomials Pure Math 0 Notes Eample : Find the area of the shaded region in factored form and as a polynomial. Shaded Area Area of Square Area of Circle Shaded Area π Area of a Circle A πr Shaded Area π (Polynomial Form) π Shaded Area (Factored Form) Radius of Circle - Homework Assignments Regular: pg. 0 # to 5 (odd), to AP: pg.0 # to (even), to Page. Copyrighted by Gabriel Tang, B.Ed., B.Sc.

9 Pure Math 0 Notes Unit : Polynomials -: Factoring b c (Leading Coefficient is ) b c What two numbers multiply to give c, but add up to be b? Eample : Completely factor the followings. a. 5 Product of b. 0 ( ) ( ) ( ) ( 5) sum of 5 Product of sum of c. a a 5 Product of d. y y (a ) (a 5) 5 5 ( y) ( y) sum of e. y y Product of f. 5w w (y ) (y ) w 5w sum of (w 5w ) (w ) (w ) g. ab ab 0a Product of sum of Rearrange in Descending Degree. Take out as common factor. () () () () 5 a (b b 0) a (b ) (b 5) Take out GCF () (5) 0 () (5) Copyrighted by Gabriel Tang, B.Ed., B.Sc. Page 9.

10 Unit : Polynomials Pure Math 0 Notes Eample : List all values of k such that the trinomial k can be factored. Product of Possible Sums for k Eample : A rectangular has an area of 9 0. a. What are the dimensions of the rectangle? b. If 5 cm, what are the actual dimensions? Factor Area for dimensions Area length width if 5 cm Area 9 0 Area 9 0 Dimensions (5 0) (5 ) Dimensions ( 0) ( ) Dimensions 5 cm cm (AP) Eample : Factor the followings. a. b. ( ) ( ) ( ) ( ) y y Assume b c as the same as b c and factor. The answer will be ( ) ( ). (y ) (y ) ( ) ( ) Do NOT Epand!! Let y ( ) ( ) ( 5) - Homework Assignments Regular: pg. #9 to 59 (odd),, 5, AP: pg. #0 to 0 (even),, 5- Page 0. Copyrighted by Gabriel Tang, B.Ed., B.Sc.

11 Pure Math 0 Notes Unit : Polynomials -9: Factoring a b c (Leading Coefficient is not, a ) For factoring trinomial with the form a b c, we will have to factor by grouping. Eample : Factor First, we look for GCF. But there is no GCF! Multiply a and c. Product of sum of Split the b term into two separate terms. ( ) ( ) Group by brackets ( ) ( ) Take out GCF for each bracket. ( ) ( ) Factor by Common Bracket! Eample : Completely factor the followings. () (9) a. y y 9 b. d d 9 9 y y y 9 d d d (y y) (y 9) (d d) (d ) sum of y (y ) (y ) d (d ) (d ) switch sign! (y ) (y ) ( sign in front (d ) (d ) of brackets) c. GCF d. m mn 9n 9 () () () () ( ) m mn mn 9n () () ( ) (m mn) (mn 9n ) [ ( sum of ) ( ) ] m (m n) n (m n) [ ( ) ( ) ] switch sign! (m n) (m n) ( sign in front ( ) ( ) of brackets) () () sum of switch sign! ( sign in front of brackets) switch sign! ( sign in front of brackets) Copyrighted by Gabriel Tang, B.Ed., B.Sc. Page.

12 Unit : Polynomials Pure Math 0 Notes Eample : List all possible values for k in 5 k so it could be factored. Product of (5) () 0 Possible Sums for k (AP) Eample : Factor the followings. a. 9 9 b. y y 9 9 () (9) () (9) y y y ( ) (9 9) ( y) ( y y ) () () () () ( ) 9 ( ) ( y) y ( y) ( ) ( 9) ( y) ( y) switch sign! ( sign in front of brackets) -9 Homework Assignments Regular: pg. 0 - # to 9 (odd), 5, 5, 55 AP: pg. 0 - # to 50 (even), 5, 5-5 Page. Copyrighted by Gabriel Tang, B.Ed., B.Sc.

13 Pure Math 0 Notes Unit : Polynomials -0: Factoring Special Quadratics Difference of Squares (Square Square) y ( y) ( y) Eample : Completely factor the followings. a. 5 b. 9 c. 00 ( 5) ( 5) (NOT Factorable Sum of Squares) ( 00) ( 0) ( 0) d. e. 9 y ( 9) ( 9) ( y) ( y) ( ) ( ) ( 9) (AP) Eample : Completely factor the followings. Look at ( ) and ( ) a. ( ) 9 Look at ( ) b. ( ) ( ) as individual items! as a single item! [ ( ) ] [ ( ) ] [ ( ) ( ) ] [ ( ) ( ) ] ( ) ( ) [ ] [5 ] ( ) (5 ) Perfect Trinomial Square a b c ( a c ) where a, c are square numbers, and b ( a )( c ) Watch Out! Subtracting a bracket! Take out negative sign from the first bracket! Eample : Epand ( ). ( ) Copyrighted by Gabriel Tang, B.Ed., B.Sc. Page.

14 Unit : Polynomials Pure Math 0 Notes Eample : Completely factor the followings. a b ( 5) ( ) (AP) Eample 5: Factor Assumes b c is the same as b c. But the answer will be in the form of ( ) ( ). ( )( 00) ( 0) Eample : List all possible values for k that can make the following polynomials as perfect squares. a. k b. k 0 5 k k 0 5 k 0 k k and ()() c. 9 5y ky The middle term of any perfect trinomial squares can have a positive or a negative numerical coefficient! 0 0 k k ( k ) 9 5y ky 9 5 ( )( k ) 5 k k Page. ( k ) -0 Homework Assignments Regular: pg. - # to (odd), 5 to 5 AP: pg. - # to (even), to 5, 59,, Copyrighted by Gabriel Tang, B.Ed., B.Sc.

15 Pure Math 0 Notes -A: Dividing Polynomials Consider 5. 5 R Original Number We can say that 5 Unit : Polynomials Divisor Quotient Divisor Original Number Divisor Quotient Or, we can say 5 () () Remainder Remainder Polynomial Function Divisor Function In general, for P() D(), we can write P( ) R Q( ) or P() D() Q() R D( ) D( ) Restriction: D() 0 Quotient Function Remainder Non-Permissible Value (NPV): - restriction on what the variable CANNOT be equal to due to the fact that the Denominator CANNOT be 0. (You can never divide by 0!) Division with Monomials Eample : Simplify the followings y a. b. y 5 ( )( ) c NPV (NPV 0) (NPV 0) Divide each term of the polynomial by the monomial. Copyrighted by Gabriel Tang, B.Ed., B.Sc. Page 5.

16 Unit : Polynomials Pure Math 0 Notes Page. Copyrighted by Gabriel Tang, B.Ed., B.Sc. Long Division to Divide Polynomials Eample : Divide R Eample : Divide R You cannot divide monomial by polynomial! Dividing by Polynomial is only possible by Long Division! OR 9 5 ( ) ( ) 5 For NPV, we let 0 NPV: OR 5 ( ) ( 9) 0 For NPV, we let 0 NPV:

17 Pure Math 0 Notes Unit : Polynomials Copyrighted by Gabriel Tang, B.Ed., B.Sc. Page. Eample : Divide 9 0 R Eample 5: Divide R 9 OR ( ) ( ) 9 For NPV, we let 0 NPV: 0 Missing Term from Decreasing Degree! OR ( ) ( ) ( ) For NPV, 0 No NPV! (cannot take square root of negative number) Missing Term from Decreasing Degree! 0 -A Homework Assignments Regular: pg. 5-5 # to (odd), 5 (a, c, e),, 0 AP: pg. 5-5 # to (even), 5 (a, c, e),,,, 5,

18 Unit : Polynomials Pure Math 0 Notes -B: Synthetic Division Only works well on divisor that is in a form of a, where Leading coefficient of Divisor is on the divisor. Eample : Divide Divisor Coefficients and constant of Polynomial Subtract numbers in each column Remainder 0 Quotient 9 (one degree less than the original polynomial) Eample : Divide 0 Quotient Remainder 9 9 Eample : Divide 5 5 Quotient 9 Remainder 9 Page. Copyrighted by Gabriel Tang, B.Ed., B.Sc.

19 Pure Math 0 Notes Unit : Polynomials The Remainder Theorem If you want to find only the remainder, you can simply substitute a from the Divisor, ( a), into the original Polynomial, P(). In general, when P( ) a, P(a) Remainder Eample : Find the remainder of the followings. a. 5 b. 0 0 P() () () 5() P() () () 0 5 Remainder 0 Remainder 9 c. 5 0 P() () () 5() 0 Remainder -B Homework Assignments Regular: pg. 5 # to ; pg. 55 Section #5 (a, b, c) AP: pg. 5 # to ; pg. 55 Section #5 (a, b, c) Copyrighted by Gabriel Tang, B.Ed., B.Sc. Page 9.

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