( ) ( ) Completing the Square. Alg 3 1 Rational Roots Solving Polynomial Equations. A Perfect Square Trinomials

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1 Alg Completing the Square A Perfect Square Trinomials (± ) ± (± ) ± 4 4 (± ) ± 6 9 (± 4) ± 8 6 (± 5) ± 5 What is the relationship between the red term and the blue term? B. Creating perfect squares.. 6 = ( ) =. 4= C. Working with coefficients. 6 = ( ) = =

2 Alg Quadratic Formula Derivation: using y = a b c Solve: 5 = 8= < 4 The Discriminant b 4ac Because it is under the radical it determines the nature of the roots.

3 Alg Algebra Assignment # Solve each of the following please. () = () 6 5= () = (4) 4 = (5) = (6) = (7) 4 < (8) 5 (9) 4 8 = () 6 5 =

4 Alg 4 Answers () ± i 5 4 () ± 6 () ± (4) ± (5), (6), 5 (7) < < (8) or (9),, i 7, i 7 (), 5

5 Alg 5 Going backwards: Writing polynomial equations from roots ROOTS = ZEROS ) roots are, - ) roots are, - ) roots are -i,

6 Alg 6 CREATING QUADRATIC EQUATIONS FROM TWO ROOTS Sum and Product Rule

7 Alg 7 Creating Equations Eercise Find the polynomial equations determined by the following roots. ), 5 4 ) ( i ),( i) ) ) 55 7 = =

8 Alg 8 Algebra Assignment # () Solve each of the following inequalities please. (a) 7 (b) ( 7)( ) ( 7)( ) (c) 4 4 < (d) 5 ( 5 )( ) (e) ( ) () Solve for please. (a) ( ) ( ) 8 = (b) = () Please write the polynomial with the following roots: (a) i, 4 (b) i, 5

9 Alg 9 Answers () (a) or (b) 5 (c) < or < < (d) < or 5 (e) 5 < or = () (a) ±,, 4 (b), 5 () (a) 6 8 4= (b) =

10 Alg Roots of Polynomials I The Remainder Theorem p( ) = 8 Calculate p() & p(). What is the remainder when p() is divided by? What is the remainder when p() is divided by? II The Factor Theorem When is a binomial (-c) a factor of a polynomial p()? III Eamples ) Is ( ) 5 a factor of f() = 8 ) Is ( ) a factor of g() = 8 5 6?

11 Alg IV Finding the roots of polynomials If ( )( 4 5 ) = 8 5, what are the roots of the polynomial? Look at the roots and look at the first and last terms of the polynomial. Any conclusions? Eamples: Find all the roots of the polynomials. ) f( ) = 4 4 ) g( ) = 6 8

12 Alg Some Shortcuts/Rules:

13 Alg Algebra Assignments # (-9) and #4 (-8) Find all roots of each of the following please. () 8 = 4 () = 5 () 4 4 = 4 (4) 6 = 4 (5) 4 = 4 (6) 5 = (7) 4 = (8) = (9) = 4 () 6 6 = () 6 6 = 4 () = () = 4 (4) 9 = 4 (5) 4 4 = ; Given i is a root 4 (6) = ; Given i is a root 4 (7) 7 5 = ; Given i is a root (8) Write an integral polynomial having the following sets of roots please (a) {,, } 4 (b) { ± i, ± }

14 Alg 4 Answers (),, (),, ± 5 (),,, ± i (4),, ± (5),, ± (6),,, 4 (7), ± (8), ± i (9), ± i () ±, ± i (), ± i (),, ± (),,, (4),, ± i (5) ± i, ± (6) ± i, ± (7) ± i, ± i (8) (a) = (b) =

15 Alg 5 Algebra Review Worksheet (Assignment #5) () Solve for please. (a) (c) 8 7 = (b) 5 9 = (d) 5 = 5 (e) 8 < (f) 8 4 () Find all roots of each of the following please. 4 (a) = (b) 4 5 = 4 (c) 7 = 4 (d) =, Given: i is a root () Write an integral polynomial having the following sets of roots please. (a),, 5 (b) { ±, ± i} (4) Find the remainder when the polynomial is divided by ( i).

16 Alg 6 Answers () (a) 5 5 ± 5 (b) 8, 4 (c) 5 ± 8 i 6 (d) or (e) 4 9 < < 4 9 (f) or () (a) { ±,, } (b) { 5, ± } (c) {,, ± i} (d) { ± i, ± 5} () (a) = 4 (b) 8 5 = (4) i

17 Alg 7 Etra Review Problems: () Solve for please. (a) = (e) 8 9 = (b) 4 = (f) = (c) 6 5 = (g) 5 4 = (d) 5 7 = (h) ( ) ( ) 5 = () Solve for please. (a) 48 6 < (c) (b) ( 5 ) ( ) (d) ( ) 6 5 () Find all values of for which 6 represents a real number.

18 Alg 8 Answers () (a) 5 4 (e) 9, (b), 4 8 (f) 6, 8 (c) ± 6 (g) 4 ± (d) 7 ± i (h), () (a) < < (c) or 4 6 (b) 5 or 5 (d) 5 or = or > () 4 < or < 4

19 Alg 9 RADICAL EQUATIONS Don t forget that radicals have restrictions!! ) 4 = ) 9 = 5 ) 5 = 4) 4 =

20 Alg () = Additional Practice Solve for please. () = () 6 = (4) 5 5 = (5) = (6) =

21 Alg () 4 () ( reject ) 4 Answers (), (4) ( reject 8 9 ) (5), (6),,

22 Alg () 8 = Algebra Assignment #6 Solve for please. () = 4 () 9 = (4) 8 = 4 (5) 5 4 = (6) = 5 (7) 4 4 = (8) 8 = 4

23 Alg 8 () ( reject ) 9 Assignment #6 Answers (), () ( reject ) (4) ( reject 5 9 ) (5) ( reject ) 4 (6) -, (7), -, (8),, 4

24 Alg 4 FRACTIONAL EQUATIONS Restrictions? Eamples ) = = 5 ) ( ) ( ) = ) 5

25 Alg 5 (4) = (5) 9 = (6) 6 4 = (7) 7 5 =

26 Alg 6 Solve each of the following for please. Algebra Assignment #7 () 5 = 5 () = () = (4) ( 4) ( 5) = ( 4) (5) = (6) 4 8 = (7) ( ) ( ) = ( ) 8 (8) =

27 Alg 7 () Assignment #7 Answers (), () (4) 9 7 (5) 4 (6) 9 (7) 5 (8) 4

28 Alg 8 Algebra Review Worksheet (Assignment #8) Solve for. () 8 5 = () = (4) = 5 () = (5) 4 = (6) 4 = (7) = (8) = (9) = () = () = 6 () 9 4 = 5

29 Alg 9 Answers () 4 () 6 () 5, (4) 5 (5), (6), 9 (7), 5 (8) 5± (9) 5 () 7 4 () 7 6 (), 8