H2adv_polynomials & Expansion Of Algebraic Expressions Worksheet

High 2 Advanced Worksheet Polynomials and Expansion of Algebraic Expressions I. Polynomials: Definitions Fill-in the blanks. 1.) A polynomial is ________________________________________________________________________. 2.) A polynomial should NOT contain terms that have: a.) variables raised to a _________________________. ex) 3x !2 and 12b !2 + 3b !1 + 4 are NOT polynomials b.) variables in the _____________________. ex)

1 1 1 1 1 + 3 and 2 + + 2 are NOT polynomials x x a ab b

c.) variables under the ________________________. ex)

xy and are x + 2 xy + y are not polynomials

3.) The degree of a term is __________________________________________________________________. ex) Given the polynomial 9xy 2 z ! 4x 3 z + 9 : a.) the degree of the first term is ______; b.) the degree of the second term is ______; c.) the degree of the third term is _______. 4.) The degree of a polynomial is _____________________________________________________________. ex) Given the polynomial 9xy 2 z ! 4x 3 z + 9 , the degree of the polynomial is ______. 5.) A polynomial in x is a mathematical expression of the form

where n is a ________________________ and the coefficients a0, a1, …, an-1, an are ____________. 6.) The leading coefficient of a polynomial in one variable is _______________________________________ _________________________________________________________________________________________. ex) The leading coefficient of the polynomial 2 + 5x 2 ! 2x 3 +

1 4 x ! x is ________. 2

II. Classifying Polynomials A. Polynomials can be classified based on the ________________. 1. monomial - _____________________________________________________________. 2. binomial - ______________________________________________________________. 3. trinomial - ______________________________________________________________. B. Polynomials can be classified based on the ________________________________. 1. linear - ________________________________________________________________. 2. quadratic - _____________________________________________________________. 3. cubic - _________________________________________________________________. 4. quartic - ________________________________________________________________. 5. quintic - ________________________________________________________________. III. Expansion of Algebraic Expressions Expand each of the following. 1.) 7 ( !5h ! 7k )

2.) !8 ( 4 p ! 3q )

3.) 5x ( 2x + 3y )

4.) !6x ( y ! 4x )

5.) 8 ( 5a ! 4 ) + 3( 2 ! 4a )

6.) 7 (12 ! 5x ) ! 3( 9 ! 7x )

7.) 5x ( !2x ! 3y ) + 2x ( !x + 3y )

8.) 4 p ( !3p + q ) ! 2 p ( !5q + p )

IV. Multiplying Binomials Expand each of the following. 1.) ( x + 5 ) ( x + 7 )

2.) ( 7 ! 2x ) ( 4 + x )

3.) ( 4x ! 5 ) ( 3x + 4 )

1 %" 1 % " 4.) $ 4 p 2 ! q ' $ 3p + q ' # 2 &# 4 &

V. Special Products Square of a Binomial ( a + b )2 = _______________________

( a ! b )2 = _____________________ Multiplying Sums and Differences

( a + b ) ( a ! b ) = __________________ Expand each of the following. 1. ( x + 5 )

3. ( 2x ! 3y )

(

2. ( a + 7 ) ( a ! 7 )

2

(

5. xy 2 ! z 2 w

)(

4. 7x + 5y 2 7x ! 5y 2

2

)

2

3 % "1 6. $ x 2 ! y' #2 5 &

2

)

VI. General Polynomial Multiplication Expand each of the following.

(

1.) ( 3 ! a ) 9 + 3a + a 2

(

)

)(

(

2.) ( x + y ) x 2 ! xy + y 2

)

3.) ( x ! 1) x 2 + x + 1 x 2 + 1

)

4.) ( 2x + 3y + 4 ) ( 2x + 3y ! 4 )

VII. Cube of a Binomial and Binomial Expansion using Pascal’s Triangle Expand each of the following 1.) ( 2x + 1)

(

3.) 5m 2 ! 4n 3

5.) ( a + b )

2.) ( a ! 2 )

3

)

3

6

3

4.) ( m + n )

5

6.) ( 2 f + 3d )

4

VIII. Reflection Question A classmate from the regular class will have a quiz on Expansions of Algebraic Expressions (all types) and he comes to you for help. In your own words, come up with a strategy so that your classmate will be able to expand any type of algebraic expression with ease and speed. (You may attach additional sheets if necessary.)