Unit 1: Polynomials

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.

Example: 5x2

Binomial: - two terms expression.

2

Example: 5x + 5x

Trinomial: - three terms expression.

Example: x2 + 5x + 6

Exponent

Variable

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

All Polynomials must have whole numbers as exponents!!

1

Example: 9 x ?1 + 12 x 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

4x

9x + 2

2

x ? 4x + 2

3

2x ? 4x2 + x + 9

4x4 ? 9x + 2

1

1

2

3

4

3

monomial

monomial

binomial

trinomial

polynomial

trinomial

Degree

Classified by Degree

0

1

1

2

3

4

constant

linear

linear

quadratic

cubic

quartic

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

Examples:

2x2y and 5x2y are like terms

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

2x2y and 5xy2 are NOT like terms

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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.

3x2 + 5x ? x2 + 4x ? 6

a.

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

=

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

2x2 + 9x ? 6

=

?x2y3 + 7x3y2

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

c.

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

=

d.

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

b.

(drop brackets and switch signs in the bracket that had

? sign in front of it)

19x2y3 + x3y2

Subtract

9x 2 + 4x

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

5x ? 7 x

2

= 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.

24 x 7 y 4 z 5

6 x 3 yz 5

c.

75a 3b 4

25a 5b 3

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

7

4

5

? 24 ?? x ?? y ?? z ?

= ? ??? 3 ???? ???? 5 ??

? 6 ?? x ?? y ?? z ?

3

4

? 75 ?? a ?? b ?

= ? ??? 5 ???? 3 ??

? 25 ?? a ?? b ?

= 21x5y6

= 4x4y3z0

= 3a?2b or

= 4x4y3

Page 2.

( z0 = 1 )

3b

a2

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 = ¦Ðr2

Inner Circle Radius = 2x

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

4x

Inner Circle Area:

A = ¦Ð (2x)2

A = ¦Ð (4x2)

A = 4¦Ðx2

Outer Circle Area:

A = ¦Ð (6x)2

A = ¦Ð (36x2)

A = 36¦Ðx2

4x

Shaded Area = 36¦Ðx2 ? 4¦Ðx2

Shaded Area = 32¦Ðx2

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

Pure Math 10 Notes

3-3: Multiplying Polynomials

To Multiply Monomials with Polynomials

Example 1: Simplify the followings.

a.

c.

3 (2x2 ? 4x + 7)

b.

2x (3x2 + 2x ? 4)

= 3 (2x2 ? 4x + 7)

= 2x (3x2 + 2x ? 4)

= 6x2 ? 12x + 21

=

3x (5x + 4) ? 4 (x2 ? 3x)

= 3x (5x + 4) ? 4 (x2 ? 3x)

= 15x2 + 12x ? 4x2 + 12x

d.

(only multiply

the brackets

right after the

monomial)

= 11x2 + 24x

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

To Multiply Polynomials with Polynomials

Example 2: Simplify the followings.

a.

(3x + 2) (4x ?3)

(x + 3) (2x2 ? 5x + 3)

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

= (x + 3) (2x2 ? 5x + 3)

= 12x2 ? 9x + 8x ? 6

= 2x3 ? 5x2 + 3x + 6x2 ? 15x + 9

=

c.

b.

12x2 ? x ? 6

3 (x + 2) (2x + 3) ? (2x ? 1) (x + 3)

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

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

= 3 (2x2 + x ? 6) ? (2x2 + 5x ? 3)

= 6x2 + 3x ? 18 ? 2x2 ? 5x + 3

= 2x3 + x2 ? 12x + 9

d.

(x2 ? 2x + 1) (3x2 + x ? 4)

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

= 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

Unit 1: Polynomials

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

5x + 4

a.

7x ? 2

b.

x+2

2x ? 1

x+1

3x + 1

x+5

Shaded Area = Big Rectangle ? Small Square

7x ? 2

= (5x + 4) (2x ? 1) ? (x + 1) (x + 1)

= (10x2 ? 5x + 8x ? 4) ? (x2 + x + x + 1)

x+2

x+2

= (10x2 + 3x ? 4) ? (x2 + 2x + 1)

= 10x2 + 3x ? 4 ? x2 ? 2x ? 1

Shaded Area = 9x2 + x ? 5

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.

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