Chapter 4

4-1 Exponents

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

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

1.
EXAMPLES x • x • x • x = x 4
2.
6 • 6 • 6 = 6 3
3.
-2 • p • q • 3 •p •q •p =
-6p 3 q 2
4.
(-2) •b • (-4) • b = 8b 2

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

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

4-2 Adding and
Subtracting Polynomials

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

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

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

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

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

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

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

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

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

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

Examples
(3x 2 y+4xy 2 – y 3 +3) +
(x 2 y+3y 3 – 4)
(-a 5 – 5ab+4b 2 – 2) –
(3a 2 – 2ab – 2b 2 – 7)

4-3 Multiplying
Monomials

RULE OF EXPONENTS
Product Rule
 a m • a n = a m + n
 x 3 • x 5 = x 8

(3n 2 )(4n 4 ) = 12n 6

RULE OF EXPONENTS
Power of a Power

(a m ) n = a mn

(x 3 ) 5 = x 15

RULE OF EXPONENTS
Power of a Product

(ab) m = a m b m

(3n 2 ) 3 = 3 3 n 6

4-5 Multiplying
Polynomials by
Monomials

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

4-6 Multiplying
Polynomials

Use the Distributive
Property

(x + 4)(x – 1)

(3x – 2)(2x 2 - 5x- 4)

(y + 2x)(x 3 – 2y 3 + 3xy 2 + x 2 y)

4-7 Transforming
Formulas

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

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

4-8 Rate-Time-
Distance Problems

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.

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

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

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

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

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

4-9 Area Problems

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

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

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

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

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

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

4-10 Problems
Without Solutions

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

Draw a Picture x + 8 x 5
5 x + 8

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

THE END