Classifying Polynomials - Grade 7 CBSE Students

1. POLYNOMIALS
POLYNOMIALS
Chapter 4

2. 4-1 Exponents

3. EXPONENTIAL FORM –
number written such
that it has a base and an
exponent
43
= 4•4 •4

4. BASE – tells what
factor is being
multiplied
EXPONENT – Tells
how many equal
factors there are

5. EXAMPLES
EXAMPLES
1.
1. x • x • x • x = x
x • x • x • x = x4
4
2.
2. 6 • 6 • 6 = 6
6 • 6 • 6 = 63
3
3.
3. -2 • p • q • 3 •p •q •p = -
-2 • p • q • 3 •p •q •p = -
6p
6p3
3
q
q2
2
4.
4. (-2) •b • (-4) • b = 8b
(-2) •b • (-4) • b = 8b2
2

6. ORDER OF OPERATIONS
ORDER OF OPERATIONS
1. Simplify expression within
grouping symbols
2. Simplify powers
3. Simplify products and
quotients in order from left to
right
4. Simplify sums and differences
in order from left to right

7. EXAMPLES
EXAMPLES
1.
1. -3
-34
4
= -(3)(3)(3)(3) = - 81
= -(3)(3)(3)(3) = - 81
2.
2. (-3)
(-3)4
4
= (-3)(-3)(-3)(-3) = 81
= (-3)(-3)(-3)(-3) = 81
3.
3. (1 + 5)
(1 + 5)2
2
= (6)
= (6)2
2
= 36
= 36
4.
4. 1 + 5
1 + 52
2
= 1 + 25 = 26
= 1 + 25 = 26

8. 4-2 Adding and
Subtracting Polynomials

9. DEFINITIONS
DEFINITIONS
Monomial
Monomial – an expression
– an expression
that is either a numeral, a
that is either a numeral, a
variable, or the product of
variable, or the product of
a numeral and one or more
a numeral and one or more
variables.
variables.
 -6xy, 14, z, 2/3r, ab
-6xy, 14, z, 2/3r, ab

10. DEFINITIONS
DEFINITIONS
Polynomial
Polynomial – an
– an
expression that is the
expression that is the
sum of monomials
sum of monomials
 14 + 2x + x
14 + 2x + x2
2
-4x
-4x

11. DEFINITIONS
DEFINITIONS
Binomial
Binomial – an expression
– an expression
that is the sum of two
that is the sum of two
monomials (has two terms)
monomials (has two terms)
 14 + 2x, x
14 + 2x, x2
2
- 4x
- 4x

12. DEFINITIONS
DEFINITIONS
Trinomial
Trinomial – an expression
– an expression
that is the sum of three
that is the sum of three
monomials (has three
monomials (has three
terms)
terms)
 14 + 2x + y, x
14 + 2x + y, x2
2
- 4x + 2
- 4x + 2

13. DEFINITIONS
DEFINITIONS
Coefficient
Coefficient – the numeral
– the numeral
preceding a variable
preceding a variable
 2x – coefficient = 2
2x – coefficient = 2

14. DEFINITIONS
DEFINITIONS
Similar terms
Similar terms – two
– two
monomials that are exactly
monomials that are exactly
alike except for their
alike except for their
coefficients
coefficients
 2x, 4x, -6x, 12x, -x
2x, 4x, -6x, 12x, -x

15. DEFINITIONS
DEFINITIONS
Simplest form
Simplest form – when no
– when no
two terms of a polynomial
two terms of a polynomial
are similar
are similar
 4x
4x3
3
– 10x
– 10x2
2
+ 2x - 1
+ 2x - 1

16. DEFINITIONS
DEFINITIONS
Degree of a variable
Degree of a variable– the
– the
number of times that the
number of times that the
variable occurs as a factor
variable occurs as a factor
in the monomial
in the monomial
 4x
4x2
2
degree of x is 2
degree of x is 2

17. DEFINITIONS
DEFINITIONS
Degree of a monomial
Degree of a monomial – the
– the
sum of the degrees of its
sum of the degrees of its
variables.
variables.
 4x
4x2
2
y degree of monomial
y degree of monomial
is 3
is 3

18. DEFINITIONS
DEFINITIONS
Degree of a polynomial
Degree of a polynomial – is
– is
the greatest of the degrees
the greatest of the degrees
of its terms after it has
of its terms after it has
been simplified.
been simplified.
 -6x
-6x3
3
+ 3x
+ 3x2
2
+ x
+ x2
2
+ 6x
+ 6x3
3
– 5
– 5

19. Examples
Examples
(3x
(3x2
2
y+4xy
y+4xy2
2
– y
– y3
3
+3) +
+3) +
(x
(x2
2
y+3y
y+3y3
3
– 4)
– 4)
(-a
(-a5
5
– 5ab+4b
– 5ab+4b2
2
– 2) –
– 2) –
(3a
(3a2
2
– 2ab – 2b
– 2ab – 2b2
2
– 7)
– 7)

20. 4-3 Multiplying
Monomials

21. RULE OF EXPONENTS
RULE OF EXPONENTS
Product Rule
Product Rule
am
• an
= am + n
x3
• x5
= x8
(3n2
)(4n4
) = 12n6

22. 4-4 Powers of
Monomials

23. RULE OF EXPONENTS
Power of a Power
(am
)n
= amn
(x3
)5
= x15

24. RULE OF EXPONENTS
Power of a Product
(ab)m
= am
bm
(3n2
)3
= 33
n6

25. 4-5 Multiplying
Polynomials by
Monomials

26. Examples – Use
Examples – Use
Distributive Property
Distributive Property
x(x + 3)
x(x + 3)
x
x2
2
+ 3x
+ 3x
4x(2x – 3)
4x(2x – 3)
8x
8x2
2
– 12x
– 12x
-2x(4x
-2x(4x2
2
– 3x + 5)
– 3x + 5)
-8x
-8x3
3
+6x
+6x2
2
– 10x
– 10x

27. 4-6 Multiplying
Polynomials

28. Use the Distributive
Use the Distributive
Property
Property
(x + 4)(x – 1)
(x + 4)(x – 1)
(3x – 2)(2x
(3x – 2)(2x2
2
- 5x- 4)
- 5x- 4)
 (y + 2x)(x
(y + 2x)(x3
3
– 2y
– 2y3
3
+ 3xy
+ 3xy2
2
+ x
+ x2
2
y)
y)

29. 4-7 Transforming
Formulas

30. Examples
Examples
C = 2
C = 2
r, solve for r
r, solve for r
c/2
c/2
 = r
= r

31. Examples
Examples
S = v/r, solve for r
S = v/r, solve for r
R = v/s
R = v/s

32. 4-8 Rate-Time-
Distance Problems

33. Example 1
Example 1
A helicopter leaves Central
Airport and flies north at 180
mi/hr. Twenty minutes later
a plane leaves the airport
and follows the helicopter at
330 mi/h. How long does it
take the plane to overtake
the helicopter.

34. Use a Chart
Rate
Rate Time
Time Distance
Distance
helicopter
helicopter 180
180 t + 1/3
t + 1/3 180(t + 1/3)
180(t + 1/3)
plane
plane 330
330 t
t 330t
330t

35. Solution
330t = 180(t + 1/3)
330t = 180(t + 1/3)
330t = 180t + 60
330t = 180t + 60
150t = 60
150t = 60
t = 2/5
t = 2/5

36. Example 2
Example 2
Bicyclists Brent and Jane started
Bicyclists Brent and Jane started
at noon from points 60 km apart
at noon from points 60 km apart
and rode toward each other,
and rode toward each other,
meeting at 1:30 PM. Brent’s
meeting at 1:30 PM. Brent’s
speed was 4 km/h greater than
speed was 4 km/h greater than
Jane’s speed. Find their
Jane’s speed. Find their
speeds.
speeds.

37. Use a Chart
Rate
Rate Time
Time Distance
Distance
Brent
Brent r + 4
r + 4 1.5
1.5 1.5(r + 4)
1.5(r + 4)
Jane
Jane r
r 1.5
1.5 1.5r
1.5r

38. Solution
Solution
1.5(r + 4) + 1.5 r = 60
1.5(r + 4) + 1.5 r = 60
1.5r + 6 + 1.5r = 60
1.5r + 6 + 1.5r = 60
3r + 6 = 60
3r + 6 = 60
3r = 54
3r = 54
r = 18
r = 18

39. 4-9 Area Problems

40. Examples
Examples
A rectangle is 5 cm longer
A rectangle is 5 cm longer
than it is wide. If its length
than it is wide. If its length
and width are both
and width are both
increased by 3 cm, its area is
increased by 3 cm, its area is
increased by 60 cm
increased by 60 cm2
2
. Find
. Find
the dimensions of the
the dimensions of the
original rectangle.
original rectangle.

41. Draw a Picture
Draw a Picture
x + 5
x
x + 8
x + 3

42. Solution
Solution
x(x+5) + 60 = (x+3)(x + 8)
x(x+5) + 60 = (x+3)(x + 8)
X
X2
2
+ 5x + 60 = x
+ 5x + 60 = x2
2
+11x + 24
+11x + 24
60 = 6x + 24
60 = 6x + 24
36 = 6x
36 = 6x
6 = x and 6 + 5 = 11
6 = x and 6 + 5 = 11

43. Example 2
Example 2
Hector made a rectangular fish
Hector made a rectangular fish
pond surrounded by a brick
pond surrounded by a brick
walk 2 m wide. He had
walk 2 m wide. He had
enough bricks for the area of
enough bricks for the area of
the walk to be 76 m
the walk to be 76 m2.
2.
Find the
Find the
dimensions of the pond if it
dimensions of the pond if it
is twice as long as it is wide.
is twice as long as it is wide.

44. Draw a Picture
Draw a Picture
2 m
2 m
2 m
2 m
2x
2x
x
x
2x + 4
2x + 4
x + 4
x + 4

45. Solution
Solution
(2x + 4)(x + 4) – (2x)(x) = 76
(2x + 4)(x + 4) – (2x)(x) = 76
2x
2x2
2
+ 8x + 4x + 16 – 2x
+ 8x + 4x + 16 – 2x2
2
= 76
= 76
12x + 16 = 76
12x + 16 = 76
-16 -16
-16 -16
12x
12x =
= 60
60
12 12
12 12
x = 5
x = 5

46. 4-10 Problems
4-10 Problems
Without Solutions
Without Solutions

47. Examples
Examples
A lawn is 8 m longer than it
A lawn is 8 m longer than it
is wide. It is surrounded
is wide. It is surrounded
by a flower bed 5 m wide.
by a flower bed 5 m wide.
Find the dimensions of
Find the dimensions of
the lawn if the area of the
the lawn if the area of the
flower bed is 140 m
flower bed is 140 m2
2

48. Draw a Picture
Draw a Picture
x + 8
x
x + 8
5
5

49. Solution
Solution
(x+10)(x+18) –x(x+8) = 140
(x+10)(x+18) –x(x+8) = 140
x
x2
2
+ 28x + 180 –x
+ 28x + 180 –x2
2
-8x = 140
-8x = 140
20x = -40
20x = -40
x = -2
x = -2
Cannot have a negative
Cannot have a negative
width
width

50. THE END
THE END