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.
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