Unit 1: Polynomials

[Pages:10]Pure Math 10 Notes

Unit 1: Polynomials

Unit 1: Polynomials

3-1: Reviewing Polynomials

Expressions: - mathematical sentences with no equal sign.

Example: 3x + 2

Equations: - mathematical sentences that are equated with an equal sign. Example: 3x + 2 = 5x + 8

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 expression.

Binomial: - two terms expression. Trinomial: - three terms expression.

Example: 5x2

Exponent

Variable

Example: 5x2 + 5x

Example: x2 + 5x + 6

Polynomial: - many terms (more than one) expression.

All Polynomials must have whole numbers as exponents!!

1

Example: 9x-1 +12x 2 is NOT a polynomial.

Degree: - the term of a polynomial that contains the largest sum of exponents Example: 9x2y3 + 4x5y2 + 3x4 Degree 7 (5 + 2 = 7)

Example 1: Fill in the table below.

Polynomial Number of Terms Classification

9

1

4x

1

9x + 2

2

x2 - 4x + 2

3

2x3 - 4x2 + x + 9

4

4x4 - 9x + 2

3

monomial monomial binomial trinomial polynomial trinomial

Degree

0 1 1 2 3 4

Classified by Degree

constant linear linear

quadratic cubic quartic

Like Terms: - terms that have the same variables and exponents.

Examples: 2x2y and 5x2y are like terms

2x2y and 5xy2 are NOT like terms

Copyrighted by Gabriel Tang, B.Ed., B.Sc.

Page 1.

Unit 1: Polynomials

Pure Math 10 Notes

To Add and Subtract Polynomials:

Combine like terms by adding or subtracting their numerical coefficients.

Example 2: Simplify the followings.

a.

3x2 + 5x - x2 + 4x - 6

b.

(9x2y3 + 4x3y2) + (3x3y2 -10x2y3)

= 3x2 + 5x - x2 + 4x - 6

= 9x2y3 + 4x3y2 + 3x3y2 -10x2y3

= 2x2 + 9x - 6

= -x2y3 + 7x3y2

c.

(9x2y3 + 4x3y2) - (3x3y2 -10x2y3)

= 9x2y3 + 4x3y2 - 3x3y2 + 10x2y3

= 19x2y3 + x3y2

(drop brackets and switch signs in the bracket that had - sign in front of it)

d. Subtract

9x2 + 4x 5x2 - 7x

This is the same as (9x2 + 4x) - (5x2 -7x)

= 9x2 + 4x - 5x2 + 7x

= 4x2 + 11x

To Multiply and Divide Monomials:

Multiply or Divide (Reduce) Numerical Coefficients. Add or Subtract exponents of the same variable according to basic exponential laws.

Example 3: Simplify the followings.

a.

(3x3y2) (7x2y4)

b.

24x7 y 4 z 5 6x3 yz5

c.

75a 3b 4 25a 5b 3

= (3)(7) (x3)(x2) (y2)(y4)

=

24 6

x 7 x3

y4 y

z5 z5

=

75 25

a 3 a5

b4 b3

= 21x5y6

= 4x4y3z0 = 4x4y3

( z0 = 1 )

=

3a-2b or

3b a2

Page 2.

Copyrighted by Gabriel Tang, B.Ed., B.Sc.

Pure Math 10 Notes

Unit 1: Polynomials

(AP) Example 4: Find the area of the following ring.

General Formula for Area of a Circle A = r2

Inner Circle Radius = 2x Outer Circle Radius = (2x + 4x) = 6x

4x

4x

Inner Circle Area:

A = (2x)2 A = (4x2) A = 4x2

Outer Circle Area:

A = (6x)2 A = (36x2) A = 36x2

Shaded Area = 36x2 - 4x2

Shaded Area = 32x2

3-1 Homework Assignment

Regular: pg. 102-103 #1 to 51, 55, 56

AP:

pg. 102-103 #1 to 51, 53-57

Copyrighted by Gabriel Tang, B.Ed., B.Sc.

Page 3.

Unit 1: Polynomials

3-3: Multiplying Polynomials

To Multiply Monomials with Polynomials

Example 1: Simplify the followings.

a. 3 (2x2 - 4x + 7)

b.

= 3 (2x2 - 4x + 7)

= 6x2 - 12x + 21

c. 3x (5x + 4) - 4 (x2 - 3x)

d.

(only multiply

= 3x (5x + 4) - 4 (x2 - 3x) the brackets

right after the

= 15x2 + 12x - 4x2 + 12x monomial)

= 11x2 + 24x

To Multiply Polynomials with Polynomials

Example 2: Simplify the followings.

a. (3x + 2) (4x -3)

b.

= (3x + 2) (4x -3)

= 12x2 - 9x + 8x - 6

= 12x2 - x - 6

Pure Math 10 Notes

2x (3x2 + 2x - 4) = 2x (3x2 + 2x - 4) = 6x3 + 4x2 - 8x 8 (a2 - 2a + 3) - 4 - (3a2 + 7)

= 8 (a2 - 2a + 3) - 4 - (3a2 + 7) = 8a2 - 16a + 24 - 4 - 3a2 - 7

= 5a2 - 16a + 13

(x + 3) (2x2 - 5x + 3) = (x + 3) (2x2 - 5x + 3) = 2x3 - 5x2 + 3x + 6x2 - 15x + 9 = 2x3 + x2 - 12x + 9

c. 3 (x + 2) (2x + 3) - (2x - 1) (x + 3)

d. (x2 - 2x + 1) (3x2 + x - 4)

= 3 (x + 2) (2x + 3) - (2x - 1) (x + 3)

= (x2 - 2x + 1) (3x2 + x - 4)

= 3 (2x2 - 3x + 4x - 6) - (2x2 + 6x - x - 3)

= 3 (2x2 + x - 6) - (2x2 + 5x - 3) = 6x2 + 3x - 18 - 2x2 - 5x + 3

= 3x4 + x3 - 4x2 -6x3 - 2x2 + 8x + 3x2 + x - 4 = 3x4 - 5x3 - 3x2 + 9x - 4

= 4x2 - 2x - 15

Page 4.

Copyrighted by Gabriel Tang, B.Ed., B.Sc.

Pure Math 10 Notes

Example 3: Find the shaded area of each of the followings.

a.

5x + 4

b.

x + 1

2x - 1 3x + 1

Unit 1: Polynomials

7x - 2

x + 2

Shaded Area = Big Rectangle - Small Square

= (5x + 4) (2x - 1) - (x + 1) (x + 1) = (10x2 - 5x + 8x - 4) - (x2 + x + x + 1) = (10x2 + 3x - 4) - (x2 + 2x + 1) = 10x2 + 3x - 4 - x2 - 2x - 1

Shaded Area = 9x2 + x - 5

x + 2

x + 5

7x - 2

x + 2

x + 5

(3x + 1) - (x + 2)

= 2x - 1 Total Area = Top Rectangle + Bottom Rectangle

= (7x -2) (x + 2) + (2x - 1) (x + 5) = (7x2 + 14x - 2x - 4) + (2x2 + 10x - x - 5) = (7x2 + 12x - 4) + (2x2 + 9x - 5) = 7x2 + 12x - 4 + 2x2 + 9x - 5

Total Area = 9x2 + 21x - 9

3-3 Homework Assignment

Regular: pg. 107-109 #1 to 77 (odd), 87, 88 AP: pg. 107-109 #2 to 84 (even) , 85, 87, 88, 91

Copyrighted by Gabriel Tang, B.Ed., B.Sc.

Page 5.

 ................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download