Download Polynomial Algebra

Lesson Objective
Understand the meaning of the words:
Be able to simplify polynomials, extend to adding, subtracting,
multiplying and dividing them
Expression
Term
Equation
Coefficient
Identity
Function
Polynomial
Root
Expression
Term
An algebraic statement made up of
variables and numbers with their
respective powers along with a some
mathematical operations of any type
Equation
An statement consisting of variables and
numbers with their respective powers that
are only multiplied or divided
Coefficient
An expression that contains an equals sign
Identity
The number part of a term
Function
Polynomial
A statement of equivalence between two
expressions that is always true for the
same inputted value
Root
Another name for a formula
Expressions, Terms, Coefficients and
Polynomials
An expression is a mathematical ‘sentence’. It contains terms
that are separated by + and – signs.
A term is a ‘word’ in the ‘sentence’.
3x
Eg 3x2 – 4y + 2x + 3xy – 4x2y + 2y +
2y
Expressions that are of the form:
a0xn + a1xn-1+ a2xn-2 + a3xn-3 + ………….. + an-1x + anx0
are called polynomials
2
2
x
y
3x2 + 2xy – 3y +
- 2xy2 +
3z
y
x2
- 0.7
How many terms?
What is the coefficient in x2?
What is the coefficient in y?
x2 y
What is the coefficient in
?
z
What is the constant term?
Which of the following expressions are polynomials?
a) 2x3 + x2 + x
2
x4
d) 3x + x-5
g)
x7
+
1/
2
x5 -
3x
b) 3x + 1
c) 6
e) sin(x) + 2
f) x3
h)
i) π
y
x2
Lesson Objectives:
Be able to simplify polynomial expressions
Be able to add, subtract and multiply polynomial expressions
Simplify:
3x + 4x2 – 2y + 3y -4x + 5x2 + 2xy – 3yx + 5x2y
Expressions are simplified by collecting together ‘like terms’
‘Like terms’ are those that contain exactly the same letters
and powers
Eg Simplify
3x + 2xy + 4x + 3yx + 2x2
Simplify
5xy2 + 3xy + 2yx + 2x2y + 3xy2
Simplify
3x + 2y – 4x + 3 – 5x2 – 3x2 + 4y + 7
Simplify:
1) 3x + 4y - 2x + 6y
2) 3 – 4x + 6 + 2x
3) 2x + 3x2 – 3x – x2
4) 2y + 3y – 4x – 5y + x
5) 3xy + 2yx + 3x – 5y – 6x – 2yx
6) 7x2 + 3x + 4x – 5x2 + 2x
7) 2x2 – 3x + 2x2y + 3xy – 4yx2 – 2x2 – 5yx
8) 3x + 2x2 – 4x + 3x2 – 2x + x
9) -5xy + 3yx + 2xy – xy + x + 3x
10) 2x2 + 3x – 4x2 – 3x + x2 – 2x + 3x2
Multiplying Polynomials
A= 2x + 3
C = 3x - 5
B = 3x2 – 2x – 5
D = 2x2 + x – 4
Find:
a) A + C
b) A + B
c) A – B
d) AC
e) A2
f) C2
g) D - B
h) 2A - C
i) BC
j) BD
k) A + BD
l) B2
m) A3
n) AD – BC
o) A ÷ D
Multiplying Polynomials
A= 2x + 3
C = 3x - 5
B = 3x2 – 2x – 5
D = 2x2 + x – 4
Find:
3x2 - 2
5x - 2
a) A + C
d) AC
b) A + B
6x2 – 9x - 15
g) D - B
-x2 + 3x + 1
e) A2
4x2 + 12x + 9
h) 2A - C
X + 11
-3x2 + 4x + 8
c) A – B
f) C2
9x2 – 30x + 25
i) BC
9x3- 21x2- 5x+ 25
j) BD
k) A4 +3 BD2
l) B2
m) A3
n) AD – BC
o) A ÷ D
15x4- x3- 24x2 + 3x +20
8x3 + 36x2 + 54x + 27
6x - x - 24x + 5x +23
-5x3 + 29x2 - 37
9x4- 12x3- 26x2 + 20x + 25
Lesson Objectives:
Dividing polynomials
Common misconceptions:
Dividing Polynomials
Find 65325 ÷ 4
What about
( 5x
.
2
 4x 1)
÷ (x  1)?
1) Find:
x 4  5 x3  2 x 2  25 x  3
x 3
4
2
2) Find the missing factor if: 4 x  11x  15x  18  (2 x  3)(.....)
3) Divide 3x 4  8 x3  10 x 2  18 x  14 by  x  2
4) Divide
x  5x  2
4
2
by
x 2  3x  1
.
We need to be able to accurately add, subtract, multiply and divide expressions:
x2+5x - 6
x+3
2x3- x2
x-1
3x2+8x+4
x2 + x
x3+ 2x2- 4x + 1
x2 + 4x + 3
x3+ 3x2+ 3x + 1
x+1
+
-
×
÷
Lesson Objective
Factorising single and double brackets
Find
1) (x + 2)(x + 5)
2) (2x + 1)(x + 4)
3) (x2 – 3)(x2 + 2x + 4)
4) (x + y)(2x – 3y + 4)
5) (2x4 – 2x2 + 3x – 5) ÷ (2x + 1)
1) 4x – 6
2) 9x + 12
3) 8x – 12
4) 12x – 9
5) 4x2 + 2x
6) 9x2 – 6x
7) 12x2 + 15x
8) 12x + 8x2
9) 20x – 15x2
10) 8xy + 12x2y
11) 8x2y – 6xy2
12) 9x2 – 27x + 12
13) x2 + 7x + 12
14) x2 + 9x + 20
15) x2 – 7x – 30
16) x2 + 3x – 18
17) x2 – 9
18) x2 + 7x – 18
19) x2 + x – 12
20) x2 - 64
21) 2x2 + 5x + 3
22) 3x2 + 5x + 2
23) 2x2 + 9x + 10
24) 5x2 + 6x + 1
25) 2x2 + 7x + 3
26) 3x2 + 9x + 20
27) 3x2 + 10x + 7
28) 7x2 + 23x + 6
29) 4x2 + 4x + 1
30) 4x2 + 5x + 1
31) 4x2 – 25
32) 16x2 - 100
Lesson Objective the Remainder Theorem
3x + 4 = 2x – 6
2(3x + 1) = 6 + 2(x – 1)
½(x + 6) = x + 1/3(2x – 5)
Page 10 and 11 Exercise Book