POLYNOMIALSPOLYNOMIALS

Chapter 4

4-1 Exponents

EXPONENTIAL FORM – number written such that it has a

base and an exponent43 = 4 •4 •4

BASE – tells what factor is

being multiplied

EXPONENT – Tells how many equal factors

there are

EXAMPLESEXAMPLES

1.1. x • x • x • x = xx • x • x • x = x44

2.2.6 • 6 • 6 = 66 • 6 • 6 = 633

3.3. -2 • p • q • 3 •p •q •p = -2 • p • q • 3 •p •q •p = -6p -6p33qq22

4.4.(-2) •b • (-4) • b = 8b(-2) •b • (-4) • b = 8b22

ORDER OF OPERATIONSORDER OF OPERATIONS1.Simplify expression

within grouping symbols2.Simplify powers3.Simplify products and

quotients in order from left to right

4.Simplify sums and differences in order from left to right

EXAMPLESEXAMPLES1.1. -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)22 = (6) = (6)22 = 36 = 36

4.4.1 + 51 + 522 = 1 + 25 = 26 = 1 + 25 = 26

4-2 Adding and Subtracting Polynomials

DEFINITIONSDEFINITIONSMonomialMonomial – an – an expression that is either expression that is either a numeral, a variable, or a numeral, a variable, or the product of a numeral the product of a numeral and one or more and one or more variables.variables.

-6xy, 14, z, 2/3r, ab-6xy, 14, z, 2/3r, ab

DEFINITIONSDEFINITIONS

PolynomialPolynomial – an – an expression that is the expression that is the sum of monomialssum of monomials

14 + 2x + x14 + 2x + x22 -4x -4x

DEFINITIONSDEFINITIONS

BinomialBinomial – an – an expression that is the expression that is the sum of two monomials sum of two monomials (has two terms)(has two terms)

14 + 2x, x14 + 2x, x22 - 4x - 4x

DEFINITIONSDEFINITIONS

TrinomialTrinomial – 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, x14 + 2x + y, x22 - 4x + - 4x + 22

DEFINITIONSDEFINITIONS

CoefficientCoefficient – the – the numeral preceding a numeral preceding a variablevariable

2x – coefficient = 22x – coefficient = 2

DEFINITIONSDEFINITIONS

Similar termsSimilar terms – two – two monomials that are monomials that are exactly alike except for exactly alike except for their coefficientstheir coefficients

2x, 4x, -6x, 12x, -x2x, 4x, -6x, 12x, -x

DEFINITIONSDEFINITIONS

Simplest formSimplest form – when no – when no two terms of a two terms of a polynomial are similarpolynomial are similar

4x4x33 – 10x – 10x22 + 2x - 1 + 2x - 1

DEFINITIONSDEFINITIONS

Degree of a variableDegree of a variable– – the number of times that the number of times that the variable occurs as a the variable occurs as a factor in the monomialfactor in the monomial

4x4x22 degree of x is 2 degree of x is 2

DEFINITIONSDEFINITIONS

Degree of a monomial Degree of a monomial – – the sum of the degrees of the sum of the degrees of its variables. its variables.

4x4x22y degree of y degree of monomial is 3monomial is 3

DEFINITIONSDEFINITIONS

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 been of its terms after it has been simplified. simplified.

-6x-6x33 + 3x + 3x22 + x + x22 + 6x + 6x33 – 5 – 5

ExamplesExamples

(3x(3x22y+4xyy+4xy22 – y – y33+3) + +3) +

(x(x22y+3yy+3y33 – 4) – 4)

(-a(-a55 – 5ab+4b – 5ab+4b22 – 2) – – 2) –

(3a(3a22 – 2ab – 2b – 2ab – 2b22 – 7) – 7)

4-3 Multiplying Monomials

RULE OF EXPONENTS RULE OF EXPONENTS Product RuleProduct Rule

am • an = am + n

x3 • x5 = x8

(3n2)(4n4) = 12n6

4-4 Powers of Monomials

RULE OF EXPONENTS

Power of a Power(am)n = amn

(x3)5 = x15

RULE OF EXPONENTS

Power of a Product(ab)m = ambm

(3n2)3 = 33n6

4-5 Multiplying Polynomials by

Monomials

Examples – Use Examples – Use Distributive Distributive

PropertyPropertyx(x + 3) x(x + 3)

xx22 + 3x + 3x4x(2x – 3) 4x(2x – 3)

8x8x22 – 12x – 12x-2x(4x-2x(4x22 – 3x + 5) – 3x + 5)

-8x-8x33+6x+6x22 – 10x – 10x

4-6 Multiplying Polynomials

Use the Distributive Use the Distributive PropertyProperty

(x + 4)(x – 1)(x + 4)(x – 1)

(3x – 2)(2x(3x – 2)(2x22- 5x- 4)- 5x- 4)

(y + 2x)(x(y + 2x)(x33 – 2y – 2y33 + 3xy + 3xy22 + x + x22y)y)

4-7 Transforming Formulas

ExamplesExamples

C = 2C = 2r, solve for rr, solve for r

c/2c/2 = r = r

ExamplesExamples

S = v/r, solve for rS = v/r, solve for r

R = v/sR = v/s

4-8 Rate-Time-Distance Problems

Example 1Example 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

RateRate TimeTime DistanceDistance

helicopterhelicopter 180180 t + 1/3t + 1/3 180(t + 1/3)180(t + 1/3)

planeplane 330330 tt 330t330t

Solution

330t = 180(t + 1/3)330t = 180(t + 1/3)

330t = 180t + 60330t = 180t + 60

150t = 60150t = 60

t = 2/5t = 2/5

Example 2Example 2

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

Use a Chart

RateRate TimeTime DistanceDistance

BrentBrent r + 4r + 4 1.51.5 1.5(r + 4)1.5(r + 4)

JaneJane rr 1.51.5 1.5r1.5r

SolutionSolution

1.5(r + 4) + 1.5 r = 601.5(r + 4) + 1.5 r = 60

1.5r + 6 + 1.5r = 601.5r + 6 + 1.5r = 60

3r + 6 = 60 3r + 6 = 60

3r = 543r = 54

r = 18r = 18

4-9 Area Problems

ExamplesExamples

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 increased by 3 cm, its area is increased by 60 cmis increased by 60 cm22. . Find the dimensions of the Find the dimensions of the original rectangle.original rectangle.

Draw a PictureDraw a Picture

x + 5

x

x + 8

x + 3

SolutionSolution

x(x+5) + 60 = (x+3)(x + 8)x(x+5) + 60 = (x+3)(x + 8)

XX22 + 5x + 60 = x + 5x + 60 = x22 +11x + +11x + 2424

60 = 6x + 2460 = 6x + 24

36 = 6x36 = 6x

6 = x and 6 + 5 = 116 = x and 6 + 5 = 11

Example 2Example 2Hector 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 mthe 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.

Draw a PictureDraw a Picture

2 m2 m

2 m2 m

2x2x

xx

2x + 42x + 4

x + 4x + 4

SolutionSolution

(2x + 4)(x + 4) – (2x)(x) = 76(2x + 4)(x + 4) – (2x)(x) = 76

2x2x22 + 8x + 4x + 16 – 2x + 8x + 4x + 16 – 2x22 = 76 = 76

12x + 16 = 7612x + 16 = 76

-16 -16-16 -16

12x12x = = 6060

12 1212 12

x = 5x = 5

4-10 Problems 4-10 Problems Without SolutionsWithout Solutions

ExamplesExamples

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 mflower bed is 140 m22

Draw a PictureDraw a Picture

x + 8

x

x + 8

5

5

SolutionSolution

(x+10)(x+18) –x(x+8) = 140(x+10)(x+18) –x(x+8) = 140xx22 + 28x + 180 –x + 28x + 180 –x22 -8x = 140 -8x = 14020x = -4020x = -40x = -2x = -2Cannot have a negative Cannot have a negative widthwidth

THE ENDTHE END