A. Find the Sum, Difference, Or Product of Two Polynomials, Then Simplify the Result

ALGEBRA p. 1

7.2 Combine and simplify algebraic expressions that contain polynomials, rational expressions, radicals, or rational exponents.

7.2 Combine and simplify algebraic expressions that contain polynomials, rational expressions, radicals, positive or negative rational exponents, and logarithmic expressions.

High School Community College EWU a. Find the sum, difference, or product of two polynomials, then simplify the result.

Core 1 SFCC – Beginning & Intermediate Algebra Basic Algebra Perform the operation and simplify: Given the polynomial 3x 2  4x  2 , add it to the 4 3 2 4 3 2 (a) 12r  7r  2r  5r  3r  2r 1 polynomial  x 2  3x 1 , and subtract it from 2x 5 . Core 2 (b) s 13 x  23x  8 Which of the following result in a value of 0? (c) 3 2 3 2 Core 3 (a)  4x  2x y 1 4x  2x y 1 (d) 2x 2  5q3x 8q hx  6x  3 5x  2 (b)  xyz 2 1  xyz 2 1

Write x  7x  7 in expanded polynomial form. SCC – Intermediate Algebra Multiply: n 2 105n  3 Find all possible sums , differences and products of the four given polynomial functions. (a) f x  4x 3  x 2  7x 17 (b) gx  3x 2  4x  7 (c) hx  5x  2 (d) jx  5x 4  3x 2

Integrated 3 Multiply 2x  3x  5 and add the product to x 2  3x  5 .

Algebra 2, CPM Bob, Kris, Janelle, and Pat are in a study team. Bob, Kris, and Janelle have algebra tiles on their desks. Bob has two x 2 -tiles, 4 x-tiles, and seven unit tiles. Kris has one x 2 tile, and five unit tiles; and Janelle has ten x-tiles, three unit tiles. Pat’s desk is empty. The team ALGEBRA p. 2

High School Community College EWU

decides to put all of the tiles from the three desks onto Pat’s desk. Write an algebraic expression for the new collection of tiles on Pat’s desk

Multiply and simplify the expression x 1x 2  2x  3

Algebra 2 Add, subtract, multiply, or divide: 5x 2  4x 3x 2  6x  7

Simplify: (a) 2b  3c3 (b)  5 f 12 4 f 3 g  2 f  (c) 3x 2  2x  4 25x 2  7

Multiply and simplify: x n 13x n  3

Algebra 2/Trigonometry Simplify: y 3 1 y 2 1 3y  7

Multiply: 4x 4  x 3  bx 2  9 and x 2  2x  3

Simplify: 2x 1x  3 3x  3

Course not indicated Find f  g , f  g , and f  g given f x  x  5 and gx  x  3 ALGEBRA p. 3

Core 4

Precalculus Given the following binomials: f x  6x 2  5x  4 and gx  2x 1 find the following: (a) f  g (b) f  g (c) f  g (d) f  g b. Factor out the greatest common factor from polynomials of any degree and from expressions involving rational exponents.

Core 1 SFCC – Beginning & Intermediate Algebra Basic Algebra Factor: Which polynomials are completely factored? (a) 125a 3 z 5  60a 4 z 4 85a 5 z 2 (a) 5(3x – 6) Core 2 (b) x(x – 3) + 5(x – 3) (b) 27m 3  9m It is not in our curriculum (c) 18x 2  x

Core 3 SCC – Intermediate Algebra 2 2 Intermediate Algebra Needs new example Factor: 10x y 15xy  35xy Rewrite the following function in an equivalent Factor by factoring out the GCF: factored form: 2x  5 3x  5 (a) kt  7t 2 14t (b) qx  x 3  5x 2  6x (made up) Algebra Concepts 2 5   Factor 2 3 2 3 . Integrated 3 (x  3)  (x  3) Factor completely: (a) 6x 2  30x  24 ALGEBRA p. 4

(b) x 4  3x 3  2x 2 Precalculus I 7 2 (c)  5x 4 15x 3 10x 2 x  1 5  x  1 5 Factor 2 x  1 5 x  2 Similar to intermediate Algebra Example

Algebra 2, CPM You should have noticed that each term in 12t 2 10t  2 is divisible by 2. That is, it has a common factor of 2. a) What is the common factor for 10x 2  25x 15 ? b) For an expression to be completely factored, each factor must have all common factors separated out. Sometimes it is easiest to do this first. Since 5 is a common factor of 10x 2  25x 15 , you can factor 10x 2  25x 15 using a special generic rectangle which is shown below. Find the length of this generic rectangle and write its area as a product of its length and width. 5 10x 2 25x 15

Factor the expression 2x  8

Algebra 2 Factor the following expressions: (a) 3x 3 15 (b) x 2  24x (c) 16x 2  x (d) xy 3  2xy 2  xy (e) 20x 3 15x 2 y 2 15xy 3 ALGEBRA p. 5

Algebra 2/Trigonometry Factor out the common factor: (a) 3xx  2 4x  2 3 1 3 (b) 4x 2x 1 2  2x2x 1 2

Course not indicated Find the zeros of f x  x 3  x 2  6x

Core 4

Precalculus Factor out the GCF: 2x 2 x 1 3xx 1

Find the zeros of the following polynomial functions: (a) f x  3x 2  27x (b) f x  3x 3  27x c. Factor quadratic polynomials with integer coefficients into a product of linear terms.

Core 2 SFCC – Beginning & Intermediate Algebra Intermediate Algebra If someone claimed that t  3 is a root of the equation Factor: Factor completely: 0  5t 2  20t  6 how could you check that claim? Test (a) a 2  9a  20 6x 2 10x  4 . your method. Is t  3 a root? (b) 6x 2 17x 12 (c) 8x 2  50 Problems similar to those shown on for SFCC and Algebra Concepts 2 EWU (d) 18k  48k  66 Factor 25x 2 10x 1 ALGEBRA p. 6

Core 3 SCC – Intermediate Algebra Precalculus Not very well done Factor: 12x 2  5x  2 Factor the following expressions: (a) x 2  5xy 14y 2 Write factored forms for each of the following 2x x quadratic function rules: (b) e  3e  2 2n n (a) f x  x 2  6x  8 (c) x  5x 14 n n (b) gx  x 2  7x 12 (d) x  5x 14x (c) hx  x 2  7x 12 (d) ix  x 2  x 12 (e) jx  x 2  5x 14 (f) kx  x 2  6x 8

Integrated 3 Factor the following: (a) x 2  6x  5 (b) x 2 144 (c) 27x 3 125y 3 (d) 5x 2  30x  25

Similar to the Algebra Concepts example

Algebra 2, CPM Factor the expression: x 2  5x  6

Algebra 2 Factor the following: (a) 4x 2 16x 10 (b) 9x 2  25 ALGEBRA p. 7

(c) x 2  5x  6

Algebra 2/Trigonometry Factor the following trinomials: (a)  9z 2  3z  2 (b) 12x 2  7x 1

Factor the trinomial by graphing: (a) 3x 2 10x  8 (b) 15x 2 11x  2

Course not indicated Factor f x  x 2  5x  6 and write f x in its linear factorization form.

Core 4 Without graphing identify all zeroes and asymptotes for each function: 2x 2  2 (a) f x  x 2 1 x 2  x 12 (b) f x  x  4

Precalculus Find the zeros of the polynomial functions (a) f x  3x 3  27x (b) f x  x 2  3x  4 ALGEBRA p. 8

Factor: (a) sin 2   3sin  2 (b) e 2 x  3e x  2 (c) x 2  3xy  2y 2 d. Simplify quotients of polynomials given in factored form, or in a form which can be factored or determine if irreducible over the real numbers.

Core 1 SFCC – Beginning & Intermediate Algebra Basic Algebra Simplify: If one of two factors of 20x 4 10x 3 100x is 10x, (a) x 2  2x 15x 2  6x  5 find the other factor using division. Core 2 2 2 It is not in our curriculum (b) a  b b  a Intermediate Algebra Core 3 6x  3 Not in curriculum; do as entry task for class SCC – Intermediate Algebra Reduce to lowest terms. 2x 2  3x  2 y  x Simplify: Weak but…write in equivalent form (in simplest 12x 2 12y 2 27 p 4  9 p 2  3 Algebra Concepts possible form): 2 9 p y 2  7y 18 Simplify y 2  3y  2 Integrated 3 2 a  9a  20 Precalculus I Simplify the following: 2 a  25 x  23  x  42 Simplify x  4 Similar to Intermediate example  

Algebra 2, CPM x 2  2x 4x 2 Divide and simplify:  x 2  4x  4 x  2 ALGEBRA p. 9

Algebra 2 Simplify: (a) 5 3  512 (b) 200x 2  50 (c) 5w 3  20w 2  3w 12 x  2x  3 (d)  xx  2 x 2  x  6 (e) x 2  4

Algebra 2/Trigonometry Write in simplest form: x 2  9 (a) x 3  x 2  9x  9 y 3  2y 2  3y (b) y 3 1

Precalculus Use synthetic division to simplify the expressions: x 3 1 (a) x 1 x 3  x 2  x 15 (b) x  3

x Find f  g if f x  and gx  x 3 x 1 ALGEBRA p. 10

x 3  x 2  x 15 Use synthetic division to simplify: x  3

a px e. Add, subtract, multiply, and divide two rational expressions of the form, where a, b, and c are real numbers such that bx  c  0 and of the form where p(x) and bx  c qx q(x) are polynomials.

Core 1 SFCC – Beginning & Intermediate Algebra Intermediate Algebra Too easy Perform the operations and simplify. Perform the operations and simplify: 1  1 2 4 (a) x x  3 (a)  Core 2 x 1 x 1 It is not in our curriculum 1 1 (b) x * x  3 2 4 (c) 3t 2  t6t 2 15t 6t 2  t 12t 2  5t  25 (b)  Core 3 x 1 x 1 Not in curriculum, do as entry task for class

SCC – Intermediate Algebra Algebra Concepts Integrated 3 2 x 3x 18 Perform the indicated operation: Add or subtract as indicated:   2 5 6 x x  3 x  3x 4x 2  x  6 3x 5 Add the following: 2   x  2 2x  4 x  3x  2 x 1 x  2

2x  6 x 2  9 Divide as indicated: 2  2 Precalculus I x  25 x 10x  25 Find the remainder when: (a) x 5  3x 2  x 1 is divided by x 3  x 4 5 5 2 Subtract:  2 (b) x  3x  x 1 is divided by x 1 x  2 x  4

Algebra 2, CPM Add or subtract and simplify (or multiply and divide; see above): ALGEBRA p. 11

2x 3x  6 (a)  x  4 x  4 8 4 (b)  x x  2

Algebra 2 Simplify the following: x 1 x 2  2x (a) 1 x 2 1 x 2  5x  6 (b) x 2  4 x  2 x 1 (c)  x 1 x  2 1 1 (d)  x x  2 x 2  7x 10 x 1 (e)  x 2  4 x  2

Algebra 2/Trigonometry Simplify the complex fraction:  x   1 (a)  2  x  2 ALGEBRA p. 12

 x 2   2  x 1  (b)  x   3  x 1 

Precalculus Divide x 4  5x 3  6x 2  x  2 by x  2 f. Simplify products and quotients of single-term expressions with rational exponents (rationalizing denominators not necessary).

Core 2 SFCC – Beginning & Intermediate Algebra Intermediate Algebra Rewrite each of the following exponential expressions Simplify the expression, eliminating negative Simplify the expression, eliminating negative in an equivalent form. For each, also state the general exponents. Assume all variables represent positive exponents. Assume all variables represent positive rule that applies: numbers. numbers. 7 z (a) 2x 2 5x 3 x 9  1  3  (a) 2 4 3 (a) 9x 8xy  z x 7   3 (b) 4  z  x 9  1  (b)    u 3 v 3   t  1 2 4 5 2 3 (b) (c) 6r  2r r r    2   1  3 2   4   uv w  (c)    x 2  (d) 3 2 35 SCC – Intermediate Algebra 1 2 Algebra Concepts Simplify: 3 3 a  a Consider the work below in simplifying. 6 5 1 Core 3     4 3  It is not in our curriculum x y Simplify  3      x 4  Integrated 3 Simplify. Write with positive exponents: ALGEBRA p. 13

2x 2 y1 3x 3 y 2  4x 2 y 5

SFCC example

Algebra 2, CPM: 1  1  3    27 

Algebra 2 Simplify: 4 2  (a) 32 5 32 5

3x  4x 2 (b) 2 x  3 3  x 4  (c)    x  6 

3 1 4  2  (d) 16x 2 4xy   

Algebra 2/Trigonometry 1 x 3  x 2 (a) 3 x 2  x 1 ALGEBRA p. 14

1 5 5 2 5x 2 (b) 3 5x 2

Precalculus Simplify the following expressions: (a) x 2  x 3 3 5 (b) x 2  x 6 2 7 (c) x 3  x 3 g. Simplify products and quotients of expressions with rational exponents and rationalize denominator when necessary.

Core 1 SFCC -- Beginning & Intermediate Algebra Precalculus I 3 7 2 1 2 1 1 5 5 3 3 6  6  x  1  x  1 Simplify z z z  z  Simplify   2 Core 2 x  1 5 x  2 It is not in our curriculum SCC – Intermediate Algebra Core 3 It is not in our curriculum

Integrated 3 1 2 x 2 y 3 z Simplify: 1 3 xy 3 z 2

Easy Problems: 4 1  2  (a)  x    ALGEBRA p. 15

1 2 (b) x y4

Algebra 2, CPM It is not in our curriculum

Algebra 2 Simplify: 3x (a) 3 4 2 3 z 2 1 2  2 3  3x y  (b)   2 1 y 4 18x 3

Algebra 2/Trigonometry 3 Simplify: 1 1 5 2  6 2

Precalculus h. Simplify rational expressions that involve complex fractions.

Core 1 SFCC -- Beginning & Intermediate Algebra Intermediate Algebra It is not in our curriculum Simplify: x 1 x  22 Simplify: Core 2 x 1 It is not in our curriculum x  2x 1 ALGEBRA p. 16

3  4 x  2 (a) Core 3 2 Algebra Concepts It is not in our curriculum 1 x  2 1 If f (x)  , find the difference quotient x 1 x x  22 f (x  h)  f (x) (b) , where h  0 and simplify your x 1 h Integrated 3 x  2x 1 answer. x  7 x 2 1 Simplify: x 2 1 Precalculus I 2 SCC – Intermediate Algebra x 14x  49 Calculate the rate of change of f (x)  x  2 between x  2 and x  a . 2 x Easy Ones-Simplify: x  2 5

Algebra 2, CPM x Simplify: 1 1 x

Algebra 2 Simplify: 3x 2  x x 2 1 (a) x x 1x 1 ALGEBRA p. 17

x 2 1 (b) x  2 3x  3 x 2  5x  6

Algebra 2/Trigonometry 1 1 2 2 2 Simplify: 4  x 2  x 4  x  2 4  x 2

Course not indicated Confirm that f and g are inverses by showing that x  3 f gx  x and g f x  x where f x  and x  2 2x  3 gx  x 1

Precalculus f x  h f x 1 Simplify: if f x  h x 1 i. Simplify logarithmic expressions.

Core 2 SFCC -- Beginning & Intermediate Algebra Precalculus I It is not in our curriculum 3log x  1 log y  3 lob z 2 Simplify : b 2 b 2 b Solve: log 8 x  2 log 8 x 1  log 8 x  4

1  x 2 x  2 Core 2 Does log 3 8  log 3  0 ?   8 Expand: ln  It is not in our curriculum  3x 1  Does log 6 60  log 6 10  1? ALGEBRA p. 18

Core 3 It is not in our curriculum SCC – Intermediate Algebra

Integrated 3 Simplify: 1 (a) log 10000

(b) log 4 64

(c) log 5 125 (d) ln 5.3

Algebra 2, CPM

Simplify: log 5 25

Algebra 2

Simplify: log 2 2  log 2 4  6 log 2 8

If c  log 3 10 and d  log 3 5 , express the following in terms of c and d:

(a) log 3 50

(b) log 3 2

(c) log 3 250

(d) log 3 500

Express as a single log: log 2 m  log 2 n  5

Algebra 2/Trigonometry ALGEBRA p. 19

Condense the following expression: 1 log x 1 2 log x 1 6 log x 2 4 4 4

Course not indicated Assuming x, y, and z are positive, use properties of logarithms to write the expression as a single logarithm: 4 logxy 3logyz

Precalculus Use properties of logarithms to write the expression as a single logarithm: (a) log x  log y (b) 1 3log x (c) 4 log y  2 log z

Condense: 21nx  3ln y  4 lnx 1 j. Factor polynomials over the complex numbers, if possible, and relate to the Fundamental Theorem of Algebra.

Core 2 SFCC -- Beginning & Intermediate Algebra Precalculus I It is not in our curriculum Find all its roots of the given equations and state the multiplicity of each root. SCC – Intermediate Algebra (a) x6 16x3  64  0 Core 2 (b) x 4  6x 3 13x 2  24x  36  0 It is not in our curriculum

Core 3 It is not in our curriculum ALGEBRA p. 20

Integrated 3 It is not in our curriculum

Algebra 2, CPM It is not in our curriculum

Algebra 2/Trigonometry Find all the zeros of the functions: (a) f x  x 3  6x 2 13x 10 (b) gx  x 4  4x 3  8x 2 16x 16

Course not indicated Find all of the zeros and write a linear factorization of the function. Explain how the solutions relate to the graph: f x  6x 4  7x 3  x 2  67x 105

Precalculus Find all of the zeros and write a linear factorization of the function f x  x 2  4x  5

Factor f x  x 5  x 4  x 3  x 2  2x  2 over the rational, real and complex numbers. How many zeros does f x have?

Find all zeros and write a linear factorization of the function f x  x 3  4x  5