1.3 Algebraic Expressions 33 22 11 Terminology. Notation. Polynomials Addition & Subtraction...

1.3 Algebraic Expressions

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2

1Terminology.

Notation.

Polynomials Addition & SubtractionMultiplication & Division

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Terminology

Set Collection of objects (elements) Usually denoted by capital letters (R, S,

T…) Elements are typically denoted with lower

case letters (a, b, c, d, …) R typically denotes the set of real numbers Z typically denotes the set of integers

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Notation

Notation or Terminology

Meaning Examples

a is an element of S

a is not an element of S

S is a subset of T every element of S is an element of T

Z is a subset of R

Constant a letter or symbol that represents a

specific element of a set

Variable a letter or symbol that represents any

element of a set

Let x denote any real number

Equal = two sets or elements of a set are identical

a=b, S=T

Not Equal two sets or elements of a set are not

identical

TSba ,

SaSa

Z3Z5

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,2,5

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Polynomials

Definition a polynomial in x is a sum of the form:

anxn + an-1xn-1 + … + a1x + a0

a monomial is an expression of the form axn, where a is a real number and n is a non-negative integer

A binomial is a sum of two monomials A trinomial is the sum of three monomials The highest value for n determines the degree of

the polynomial The coefficient, a, associated with the highest

value of n is the leading coefficient

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Polynomials (cont.)

Example Leading Coefficient Degree

3x4 + 5x3 + (-7)x + 4 3 4

x8 + 9x2 + (-2)x 1 8

-5x2 + 1 -5 2

7x + 2 7 1

8 8 0

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Polynomials (cont.)

Adding

Subtracting

10535

354752

)354()752(

23

2323

2323

xxx

xxxxx

xxxxx

4573

354752

)354(1)752(

)354()752(

23

2323

2323

2323

xxx

xxxxx

xxxxx

xxxxx

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Polynomials (cont.)

Multiplying Polynomials

417145102

41285151032

)132(4)132(5)132(

)132)(45(

2345

324235

3332

32

xxxxx

xxxxxxxx

xxxxxxxx

xxxx

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Polynomials (cont.)

Special Product Formulas Example

(x + y)(x - y) = x2 – y2 (2a + 3)(2a – 3) = (2a)2 – 32

= 4a2 - 9

(x ±y)2 = x2 ± 2xy + y2 (2a – 3)2 = (2a)2 – 2(2a)(3)+32

= 4a2 - 12a + 9

(x ± y)3 = x3 ± 3x2y + 3xy2 ± y3 (2a + 3)3

= (2a)3 + 3(2a)2(3) + 3(2a)(3)2+(3)3

=8a3 + 36a2 + 54a + 27

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Polynomials (cont.)

Dividing a Polynomial by a Binomial

523

2

10

2

4

2

6

2

1046

22

2332

2332

yxxy

xy

xy

xy

yx

xy

yx

xy

xyyxyx

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Practice Problems

Page 43Problems 1-44

1.3 Algebraic Expressions 33 22 11 Terminology. Notation. Polynomials Addition & Subtraction...

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