Lecture 281 Unit 4 Lecture 28 Introduction to Quadratics Introduction to Quadratics

Lecture 28Lecture 28 11

Unit 4 Lecture 28Unit 4 Lecture 28Introduction to QuadraticsIntroduction to Quadratics

Introduction to QuadraticsIntroduction to Quadratics

Lecture 28Lecture 28 22

ObjectivesObjectives• Evaluate quadratic expressionsEvaluate quadratic expressions• Identify the degree of a polynomialIdentify the degree of a polynomial• Determine the number of terms in a polynomialDetermine the number of terms in a polynomial• Add and subtract polynomialsAdd and subtract polynomials• Multiply binomials using FOILMultiply binomials using FOIL

Lecture 28Lecture 28 33

WT shoots an arrow straight up with an initial WT shoots an arrow straight up with an initial velocity of 160 feet per second. The height (in feet) velocity of 160 feet per second. The height (in feet) of the arrow is given by the equation:of the arrow is given by the equation: h = -16t h = -16t22 + 160 t + 160 tt is the number of seconds the arrow is in the air.t is the number of seconds the arrow is in the air.Find the height of the arrow.Find the height of the arrow.

tt h=-16th=-16t22 + 160 t + 160 t

22

55

88

1010

Lecture 28Lecture 28 44

WT shoots an arrow straight up with an initial WT shoots an arrow straight up with an initial velocity of 160 feet per second. The height (in feet) velocity of 160 feet per second. The height (in feet) of the arrow is given by the equation:of the arrow is given by the equation: h=-16t h=-16t22 + 160 t + 160 tt is the number of seconds the arrow is in the air.t is the number of seconds the arrow is in the air.Find the height of the arrow.Find the height of the arrow.

tt h=-16th=-16t22 + 160 t + 160 t

22 -16(-16(22))22 + 160( + 160(22) =) =

55

88

1010

Lecture 28Lecture 28 55

WT shoots an arrow straight up with an initial WT shoots an arrow straight up with an initial velocity of 160 feet per second. The height (in feet) velocity of 160 feet per second. The height (in feet) of the arrow is given by the equation:of the arrow is given by the equation: h=-16t h=-16t22 + 160 t + 160 tt is the number of seconds the arrow is in the air.t is the number of seconds the arrow is in the air.Find the height of the arrow.Find the height of the arrow.

tt h=-16th=-16t22 + 160 t + 160 t

22 -16(-16(22))22 + 160( + 160(22) = -64+320 = 256) = -64+320 = 256

55

88

1010

Lecture 28Lecture 28 66

WT shoots an arrow straight up with an initial WT shoots an arrow straight up with an initial velocity of 160 feet per second. The height (in feet) velocity of 160 feet per second. The height (in feet) of the arrow is given by the equation:of the arrow is given by the equation: h=-16t h=-16t22 + 160 t + 160 tt is the number of seconds the arrow is in the air.t is the number of seconds the arrow is in the air.Find the height of the arrow.Find the height of the arrow.

tt h=-16th=-16t22 + 160 t + 160 t

22 -16(-16(22))22 + 160( + 160(22) = -64+320 = 256) = -64+320 = 256

55 -16(-16(55))22 + 160( + 160(55) =) =

88

1010

Lecture 28Lecture 28 77

WT shoots an arrow straight up with an initial WT shoots an arrow straight up with an initial velocity of 160 feet per second. The height (in feet) velocity of 160 feet per second. The height (in feet) of the arrow is given by the equation:of the arrow is given by the equation: h=-16t h=-16t22 + 160 t + 160 tt is the number of seconds the arrow is in the air.t is the number of seconds the arrow is in the air.Find the height of the arrow.Find the height of the arrow.

tt h=-16th=-16t22 + 160 t + 160 t

22 -16(-16(22))22 + 160( + 160(22) = -64+320 = 256) = -64+320 = 256

55 -16(-16(55))22 + 160( + 160(55) = -400+800 = 400) = -400+800 = 400

88

1010

Lecture 28Lecture 28 88

WT shoots an arrow straight up with an initial WT shoots an arrow straight up with an initial velocity of 160 feet per second. The height (in feet) velocity of 160 feet per second. The height (in feet) of the arrow is given by the equation:of the arrow is given by the equation: h=-16t h=-16t22 + 160 t + 160 tt is the number of seconds the arrow is in the air.t is the number of seconds the arrow is in the air.Find the height of the arrow.Find the height of the arrow.

tt h=-16th=-16t22 + 160 t + 160 t

22 -16(-16(22))22 + 160( + 160(22) = -64+320 = 256) = -64+320 = 256

55 -16(-16(55))22 + 160( + 160(55) = -400+800 = 400) = -400+800 = 400

88 -16(-16(88))22 + 160( + 160(88) =) =

1010

Lecture 28Lecture 28 99

WT shoots an arrow straight up with an initial WT shoots an arrow straight up with an initial velocity of 160 feet per second. The height (in feet) velocity of 160 feet per second. The height (in feet) of the arrow is given by the equation:of the arrow is given by the equation: h=-16t h=-16t22 + 160 t + 160 tt is the number of seconds the arrow is in the air.t is the number of seconds the arrow is in the air.Find the height of the arrow.Find the height of the arrow.

tt h=-16th=-16t22 + 160 t + 160 t

22 -16(-16(22))22 + 160( + 160(22) = -64+320 = 256) = -64+320 = 256

55 -16(-16(55))22 + 160( + 160(55) = -400+800 = 400) = -400+800 = 400

88 -16(-16(88))22 + 160( + 160(88) = -1024+1280 = 256) = -1024+1280 = 256

1010

Lecture 28Lecture 28 1010

WT shoots an arrow straight up with an initial WT shoots an arrow straight up with an initial velocity of 160 feet per second. The height (in feet) velocity of 160 feet per second. The height (in feet) of the arrow is given by the equation:of the arrow is given by the equation: h=-16t h=-16t22 + 160 t + 160 tt is the number of seconds the arrow is in the air.t is the number of seconds the arrow is in the air.Find the height of the arrow.Find the height of the arrow.

tt h=-16th=-16t22 + 160 t + 160 t

22 -16(-16(22))22 + 160( + 160(22) = -64+320 = 256) = -64+320 = 256

55 -16(-16(55))22 + 160( + 160(55) = -400+800 = 400) = -400+800 = 400

88 -16(-16(88))22 + 160( + 160(88) = -1024+1280 = 256) = -1024+1280 = 256

1010 -16(-16(1010))22 + 160( + 160(1010) =) =

Lecture 28Lecture 28 1111

WT shoots an arrow straight up with an initial WT shoots an arrow straight up with an initial velocity of 160 feet per second. The height (in feet) velocity of 160 feet per second. The height (in feet) of the arrow is given by the equation:of the arrow is given by the equation: h=-16t h=-16t22 + 160 t + 160 tt is the number of seconds the arrow is in the air.t is the number of seconds the arrow is in the air.Find the height of the arrow.Find the height of the arrow.

tt h=-16th=-16t22 + 160 t + 160 t

22 -16(-16(22))22 + 160( + 160(22) = -64+320 = 256) = -64+320 = 256

55 -16(-16(55))22 + 160( + 160(55) = -400+800 = 400) = -400+800 = 400

88 -16(-16(88))22 + 160( + 160(88) = -1024+1280 = 256) = -1024+1280 = 256

1010 -16(-16(1010))22 + 160( + 160(1010) = -1600+1600 = 0) = -1600+1600 = 0

Lecture 28Lecture 28 1212

WT shoots an arrow straight up with an initial WT shoots an arrow straight up with an initial velocity of 160 feet per second. The height (in feet) velocity of 160 feet per second. The height (in feet) of the arrow is given by the equation:of the arrow is given by the equation: h=-16t h=-16t22 + 160 t + 160 tt is the number of seconds the arrow is in the air.t is the number of seconds the arrow is in the air.When will the arrow reach its maximum height?When will the arrow reach its maximum height?

tt hh22 256256

55 400400

88 256256

1010 00

Lecture 28Lecture 28 1313

WT shoots an arrow straight up with an initial WT shoots an arrow straight up with an initial velocity of 160 feet per second. The height (in feet) velocity of 160 feet per second. The height (in feet) of the arrow is given by the equation:of the arrow is given by the equation: h=-16t h=-16t22 + 160 t + 160 tt is the number of seconds the arrow is in the air.t is the number of seconds the arrow is in the air.

When will the When will the arrow reach its arrow reach its maximum height?maximum height?

tt hh22 256256

55 400400

88 256256

1010 00

When When time = 5 sectime = 5 sec, , height = 400 feetheight = 400 feet..

Lecture 28Lecture 28 1414

WT shoots an arrow straight up with an initial WT shoots an arrow straight up with an initial velocity of 160 feet per second. The height (in feet) velocity of 160 feet per second. The height (in feet) of the arrow is given by the equation:of the arrow is given by the equation: h=-16t h=-16t22 + 160 t + 160 tt is the number of seconds the arrow is in the air.t is the number of seconds the arrow is in the air.

When will the When will the arrow hit the arrow hit the ground?ground?

tt hh22 256256

55 400400

88 256256

1010 00

Lecture 28Lecture 28 1515

WT shoots an arrow straight up with an initial WT shoots an arrow straight up with an initial velocity of 160 feet per second. The height (in feet) velocity of 160 feet per second. The height (in feet) of the arrow is given by the equation:of the arrow is given by the equation: h=-16t h=-16t22 + 160 t + 160 tt is the number of seconds the arrow is in the air.t is the number of seconds the arrow is in the air.

When will the When will the arrow hit the arrow hit the ground?ground?

tt hh22 256256

55 400400

88 256256

1010 00

When When time = 10 sectime = 10 sec, , height = 0 feetheight = 0 feet..

Lecture 28Lecture 28 1616

h = -16th = -16t22+ 160t+ 160t

100

400

200

300

2 4 6 Time

tt hh22 25625655 40040088 256256

1010 00

8 10

Height

Lecture 28Lecture 28 1717

Term:Term:DefinitionsDefinitions

Lecture 28Lecture 28 1818

Term:Term: A number or the product of a A number or the product of a number and a variable number and a variable raised to a power.raised to a power.

DefinitionsDefinitions

Lecture 28Lecture 28 1919

Term:Term: A number or the product of a A number or the product of a number and a variable number and a variable raised to a power.raised to a power.

Example:Example: 2x2x33, 3, 4x, 5x, 3, 4x, 5x66, y, x, y, x

DefinitionsDefinitions

Lecture 28Lecture 28 2020

Term:Term: A number or the product of a A number or the product of a number and a variable number and a variable raised to a power.raised to a power.

Example:Example: 2x2x33, 3, 4x, 5x, 3, 4x, 5x66, y, x, y, x

Like TermsLike Terms

DefinitionsDefinitions

Lecture 28Lecture 28 2121

Term:Term: A number or the product of a A number or the product of a number and a variable number and a variable raised to a power.raised to a power.

Example:Example: 2x2x33, 3, 4x, 5x, 3, 4x, 5x66, y, x, y, x

Like TermsLike Terms Terms with the same variables Terms with the same variables raised to exactly the same raised to exactly the same powers.powers.

DefinitionsDefinitions

Lecture 28Lecture 28 2222

Term:Term: A number or the product of a A number or the product of a number and a variable number and a variable raised to a power.raised to a power.

Example:Example: 2x2x33, 3, 4x, 5x, 3, 4x, 5x66, y, x, y, x

Like TermsLike Terms Terms with the same variables Terms with the same variables raised to exactly the same raised to exactly the same powers.powers.

Example:Example: 3x3x22 , 4x , 4x22

DefinitionsDefinitions

Lecture 28Lecture 28 2323

How do we combine like terms?How do we combine like terms?

Lecture 28Lecture 28 2424

How do we combine like terms?How do we combine like terms?

Add the numerical coefficients and Add the numerical coefficients and multiply the result by the common multiply the result by the common variable factor.variable factor.

Lecture 28Lecture 28 2525

How do we combine like terms?How do we combine like terms?

Add the numerical coefficients and Add the numerical coefficients and multiply the result by the common multiply the result by the common variable factor.variable factor.

-8x-8x22 + 2x – 8 – 6x + 2x – 8 – 6x22 + 10x + 2 + 10x + 2

Lecture 28Lecture 28 2626

How do we combine like terms?How do we combine like terms?

Add the numerical coefficients and Add the numerical coefficients and multiply the result by the common multiply the result by the common variable factor.variable factor.

-8x-8x22 + 2x – 8 – 6x + 2x – 8 – 6x22 + 10x + 2 + 10x + 2-8x-8x22 ++ 2x2x – 8– 8 – 6x– 6x22 + 10x+ 10x + 2+ 2

Lecture 28Lecture 28 2727

How do we combine like terms?How do we combine like terms?

Add the numerical coefficients and Add the numerical coefficients and multiply the result by the common multiply the result by the common variable factor.variable factor.

-8x-8x22 + 2x – 8 – 6x + 2x – 8 – 6x22 + 10x + 2 + 10x + 2-8x-8x22 ++ 2x2x – 8– 8 – 6x– 6x22 + 10x+ 10x + 2+ 2

-14x-14x22 + 12x+ 12x – 6– 6

Lecture 28Lecture 28 2828

Simplify:Simplify:

2(3x2(3x22 + 5x – 10) – 4(x + 5x – 10) – 4(x22 – 6x + 3) – 6x + 3)

Lecture 28Lecture 28 2929

Simplify:Simplify:

2(3x2(3x22 + 5x – 10) – 4(x + 5x – 10) – 4(x22 – 6x + 3) – 6x + 3)

6x6x22 + 10x – 20 – 4x + 10x – 20 – 4x22 + 24x - 12 + 24x - 12

Lecture 28Lecture 28 3030

Simplify:Simplify:

2(3x2(3x22 + 5x – 10) – 4(x + 5x – 10) – 4(x22 – 6x + 3) – 6x + 3)

6x6x22 ++ 10x10x – 20– 20 – 4x– 4x22 + 24x+ 24x – 12– 126x6x22 + 10x – 20 – 4x + 10x – 20 – 4x22 + 24x - 12 + 24x - 12

Lecture 28Lecture 28 3131

Simplify:Simplify:

2(3x2(3x22 + 5x – 10) – 4(x + 5x – 10) – 4(x22 – 6x + 3) – 6x + 3)

6x6x22 ++ 10x10x – 20– 20 – 4x– 4x22 + 24x+ 24x – 12– 12

2x2x22 + 34x+ 34x – 32– 32

6x6x22 + 10x – 20 – 4x + 10x – 20 – 4x22 + 24x - 12 + 24x - 12

Lecture 28Lecture 28 3232

Simplify:Simplify:

5(2x5(2x22 + 5) – (8x + 5) – (8x22 + 2x – 4) + 2x – 4)

Lecture 28Lecture 28 3333

Simplify:Simplify:

5(2x5(2x22 + 5) – (8x + 5) – (8x22 + 2x – 4) + 2x – 4)

10x10x22 + 25 – 8x + 25 – 8x22 – 2x + 4 – 2x + 4

Lecture 28Lecture 28 3434

Simplify:Simplify:

5(2x5(2x22 + 5) – (8x + 5) – (8x22 + 2x – 4) + 2x – 4)

10x10x22 + 25 – 8x + 25 – 8x22 – 2x + 4 – 2x + 410x10x22 + 25+ 25 – 8x– 8x22 – 2x– 2x + 4+ 4

Lecture 28Lecture 28 3535

Simplify:Simplify:

5(2x5(2x22 + 5) – (8x + 5) – (8x22 + 2x – 4) + 2x – 4)

2x2x2 2 – 2x– 2x + 29+ 29

10x10x22 + 25 – 8x + 25 – 8x22 – 2x + 4 – 2x + 410x10x22 + 25+ 25 – 8x– 8x22 – 2x– 2x + 4+ 4

Lecture 28Lecture 28 3636

The equation for profit is, The equation for profit is, Profit = Profit = RevenueRevenue - - CostCost

Revenue:Revenue: R = -5xR = -5x22 + 17x + 17x

Cost:Cost: C = 3xC = 3x22 - 27x + 40 - 27x + 40

Lecture 28Lecture 28 3737

The equation for profit is, The equation for profit is, Profit = Profit = RevenueRevenue - - CostCost

Revenue:Revenue: R = -5xR = -5x22 + 17x + 17x

P = (Revenue) – (Cost)P = (Revenue) – (Cost)

Cost:Cost: C = 3xC = 3x22 - 27x + 40 - 27x + 40

Lecture 28Lecture 28 3838

The equation for profit is, The equation for profit is, Profit = Profit = RevenueRevenue - - CostCost

Revenue:Revenue: R = -5xR = -5x22 + 17x + 17x

P = (-5xP = (-5x22 + 17x) – + 17x) – (Cost)(Cost)

Cost:Cost: C = 3xC = 3x22 - 27x + 40 - 27x + 40

Lecture 28Lecture 28 3939

The equation for profit is, The equation for profit is, Profit = Profit = RevenueRevenue - - CostCost

Revenue:Revenue: R = -5xR = -5x22 + 17x + 17x

P = (-5xP = (-5x22 + 17x) – ( + 17x) – (3x3x22 - 27x + 40 - 27x + 40))

Cost:Cost: C = 3xC = 3x22 - 27x + 40 - 27x + 40

Lecture 28Lecture 28 4040

The equation for profit is, The equation for profit is, Profit = Profit = RevenueRevenue - - CostCost

Revenue:Revenue: R = -5xR = -5x22 + 17x + 17x

P = (-5xP = (-5x22 + 17x) – ( + 17x) – (3x3x22 - 27x + 40 - 27x + 40))

Cost:Cost: C = 3xC = 3x22 - 27x + 40 - 27x + 40

P = – 5xP = – 5x22 + 17x + 17x – 3x– 3x22 + 27x – 40 + 27x – 40

Lecture 28Lecture 28 4141

The equation for profit is, The equation for profit is, Profit = Profit = RevenueRevenue - - CostCost

Revenue:Revenue: R = -5xR = -5x22 + 17x + 17x

P = P = -8x-8x22

P = (-5xP = (-5x22 + 17x) – ( + 17x) – (3x3x22 - 27x + 40 - 27x + 40))

Cost:Cost: C = 3xC = 3x22 - 27x + 40 - 27x + 40

P = – 5xP = – 5x22 + 17x + 17x – 3x– 3x22 + 27x – 40 + 27x – 40

Lecture 28Lecture 28 4242

The equation for profit is, The equation for profit is, Profit = Profit = RevenueRevenue - - CostCost

Revenue:Revenue: R = -5xR = -5x22 + 17x + 17x

P = P = - 8x- 8x2 2 + 44x+ 44x

P = (-5xP = (-5x22 + 17x) – ( + 17x) – (3x3x22 - 27x + 40 - 27x + 40))

Cost:Cost: C = 3xC = 3x22 - 27x + 40 - 27x + 40

P = – 5xP = – 5x22 + 17x + 17x – 3x– 3x22 + 27x – 40 + 27x – 40

Lecture 28Lecture 28 4343

The equation for profit is, The equation for profit is, Profit = Profit = RevenueRevenue - - CostCost

Revenue:Revenue: R = -5xR = -5x22 + 17x + 17x

P = P = - 8x- 8x2 2 + 44x+ 44x - 40- 40

P = (-5xP = (-5x22 + 17x) – ( + 17x) – (3x3x22 - 27x + 40 - 27x + 40))

Cost:Cost: C = 3xC = 3x22 - 27x + 40 - 27x + 40

P = – 5xP = – 5x22 + 17x + 17x – 3x– 3x22 + 27x – 40 + 27x – 40

Lecture 28Lecture 28 4444

Polynomial:Polynomial:DefinitionsDefinitions

Lecture 28Lecture 28 4545

Polynomial:Polynomial: A term or a finite sum of terms A term or a finite sum of terms in which variables may in which variables may appear in the numerator appear in the numerator raised to whole number raised to whole number powers only.powers only.

DefinitionsDefinitions

Lecture 28Lecture 28 4646

Polynomial:Polynomial: A term or a finite sum of terms A term or a finite sum of terms in which variables may in which variables may appear in the numerator appear in the numerator raised to whole number raised to whole number powers only.powers only.

Examples:Examples: 2x2x33

DefinitionsDefinitions

Lecture 28Lecture 28 4747

Polynomial:Polynomial: A term or a finite sum of terms A term or a finite sum of terms in which variables may in which variables may appear in the numerator appear in the numerator raised to whole number raised to whole number powers only.powers only.

Examples:Examples: 2x2x33

3 +x3 +x

DefinitionsDefinitions

Lecture 28Lecture 28 4848

Polynomial:Polynomial: A term or a finite sum of terms A term or a finite sum of terms in which variables may in which variables may appear in the numerator appear in the numerator raised to whole number raised to whole number powers only.powers only.

Examples:Examples: 2x2x33

3 +x3 +x7+x+5x7+x+5x6 6

DefinitionsDefinitions

Lecture 28Lecture 28 4949

Polynomial:Polynomial: A term or a finite sum of terms A term or a finite sum of terms in which variables may in which variables may appear in the numerator appear in the numerator raised to whole number raised to whole number powers only.powers only.

Examples:Examples: 2x2x33

3 +x3 +x7+x+5x7+x+5x6 6

xx

DefinitionsDefinitions

Lecture 28Lecture 28 5050

Polynomials:Polynomials: Nonpolynomials:Nonpolynomials:

DefinitionsDefinitions

2

3 12 3 5

xx x

Lecture 28Lecture 28 5151

Polynomials:Polynomials: Nonpolynomials:Nonpolynomials:

DefinitionsDefinitions

2

3 12 3 5

xx x

3 17x

Lecture 28Lecture 28 5252

Polynomials:Polynomials: Nonpolynomials:Nonpolynomials:DefinitionsDefinitions

2

3 12 3 5

xx x

2x

3 17x

Lecture 28Lecture 28 5353

Polynomials:Polynomials: Nonpolynomials:Nonpolynomials:DefinitionsDefinitions

2

3 12 3 5

xx x

2x23

3 17x

Lecture 28Lecture 28 5454

Polynomials:Polynomials: Nonpolynomials:Nonpolynomials:DefinitionsDefinitions

2

3 12 3 5

xx x

2x

3 42 3xx yy

23

3 17x

Lecture 28Lecture 28 5555

Polynomials:Polynomials: Nonpolynomials:Nonpolynomials:DefinitionsDefinitions

2

3 12 3 5

xx x

2x

3 42 3xx yy

23

2 23 2x xy y

3 17x

Lecture 28Lecture 28 5656

Polynomials:Polynomials: Nonpolynomials:Nonpolynomials:DefinitionsDefinitions

2

3 12 3 5

xx x

2x

3 42 3xx yy

1x

23

2 23 2x xy y

3 17x

Lecture 28Lecture 28 5757

Polynomials:Polynomials: Nonpolynomials:Nonpolynomials:DefinitionsDefinitions

2

3 12 3 5

xx x

2x

3 42 3xx yy

1x

23

2 23 2x xy y

2 21 73

x xy y

3 17x

Lecture 28Lecture 28 5858

Monomial:Monomial:Examples:Examples:

Binomial:Binomial:Examples:Examples:

Trinomial:Trinomial:Examples:Examples:

DefinitionsDefinitions

Lecture 28Lecture 28 5959

Monomial:Monomial: A polynomial with exactly 1 term.A polynomial with exactly 1 term.

Examples:Examples:

Binomial:Binomial:Examples:Examples:

Trinomial:Trinomial:Examples:Examples:

DefinitionsDefinitions

Lecture 28Lecture 28 6060

Monomial:Monomial: A polynomial with exactly 1 term.A polynomial with exactly 1 term.

Examples:Examples: 2x2x33, x, 7, y, 5x, x, 7, y, 5x66

Binomial:Binomial:Examples:Examples:

Trinomial:Trinomial:Examples:Examples:

DefinitionsDefinitions

Lecture 28Lecture 28 6161

Monomial:Monomial: A polynomial with exactly 1 term.A polynomial with exactly 1 term.

Examples:Examples: 2x2x33, x, 7, y, 5x, x, 7, y, 5x66

Binomial:Binomial: A polynomial with exactly 2 terms.A polynomial with exactly 2 terms.

Examples:Examples:

Trinomial:Trinomial:Examples:Examples:

DefinitionsDefinitions

Lecture 28Lecture 28 6262

Monomial:Monomial: A polynomial with exactly 1 term.A polynomial with exactly 1 term.

Examples:Examples: 2x2x33, x, 7, y, 5x, x, 7, y, 5x66

Binomial:Binomial: A polynomial with exactly 2 terms.A polynomial with exactly 2 terms.

Examples:Examples: 2x2x33+9x, 3 +x, 7+5x+9x, 3 +x, 7+5x66

Trinomial:Trinomial:Examples:Examples:

DefinitionsDefinitions

Lecture 28Lecture 28 6363

Monomial:Monomial: A polynomial with exactly 1 term.A polynomial with exactly 1 term.

Examples:Examples: 2x2x33, x, 7, y, 5x, x, 7, y, 5x66

Binomial:Binomial: A polynomial with exactly 2 terms.A polynomial with exactly 2 terms.

Examples:Examples: 2x2x33+9x, 3 +x, 7+5x+9x, 3 +x, 7+5x66

Trinomial:Trinomial: A polynomial with exactly 3 terms.A polynomial with exactly 3 terms.

Examples:Examples:

DefinitionsDefinitions

Lecture 28Lecture 28 6464

Monomial:Monomial: A polynomial with exactly 1 term.A polynomial with exactly 1 term.

Examples:Examples: 2x2x33, x, 7, y, 5x, x, 7, y, 5x66

Binomial:Binomial: A polynomial with exactly 2 terms.A polynomial with exactly 2 terms.

Examples:Examples: 2x2x33+9x, 3 +x, 7+5x+9x, 3 +x, 7+5x66

Trinomial:Trinomial: A polynomial with exactly 3 terms.A polynomial with exactly 3 terms.

Examples:Examples: 3x3x22+6x+2 , 7+5x+6x+2 , 7+5x33+4x+4x22

DefinitionsDefinitions

Lecture 28Lecture 28 6565

Degree of a term:Degree of a term:

Examples:Examples:

DefinitionsDefinitions

Lecture 28Lecture 28 6666

Degree of a term:Degree of a term: The sum of the exponents on The sum of the exponents on the variables in the term.the variables in the term.

Examples:Examples:

DefinitionsDefinitions

Lecture 28Lecture 28 6767

Degree of a term:Degree of a term: The sum of the exponents on The sum of the exponents on the variables in the term.the variables in the term.

Examples:Examples:

DefinitionsDefinitions

TermTerm DegreeDegree

2x2x33

Lecture 28Lecture 28 6868

Degree of a term:Degree of a term: The sum of the exponents on The sum of the exponents on the variables in the term.the variables in the term.

Examples:Examples:

DefinitionsDefinitions

TermTerm DegreeDegree

2x2x33 33

Lecture 28Lecture 28 6969

Degree of a term:Degree of a term: The sum of the exponents on The sum of the exponents on the variables in the term.the variables in the term.

Examples:Examples:

DefinitionsDefinitions

TermTerm DegreeDegree

2x2x33 335x5x66

Lecture 28Lecture 28 7070

Degree of a term:Degree of a term: The sum of the exponents on The sum of the exponents on the variables in the term.the variables in the term.

Examples:Examples:

DefinitionsDefinitions

TermTerm DegreeDegree

2x2x33 335x5x66 66

Lecture 28Lecture 28 7171

Degree of a term:Degree of a term: The sum of the exponents on The sum of the exponents on the variables in the term.the variables in the term.

Examples:Examples:

DefinitionsDefinitions

TermTerm DegreeDegree

2x2x33 335x5x66 66xx

Lecture 28Lecture 28 7272

Degree of a term:Degree of a term: The sum of the exponents on The sum of the exponents on the variables in the term.the variables in the term.

Examples:Examples:

DefinitionsDefinitions

TermTerm DegreeDegree

2x2x33 335x5x66 66xx 11

Lecture 28Lecture 28 7373

Degree of a term:Degree of a term: The sum of the exponents on The sum of the exponents on the variables in the term.the variables in the term.

Examples:Examples:

DefinitionsDefinitions

TermTerm DegreeDegree

2x2x33 335x5x66 66xx 11xyxy

Lecture 28Lecture 28 7474

Degree of a term:Degree of a term: The sum of the exponents on The sum of the exponents on the variables in the term.the variables in the term.

Examples:Examples:

DefinitionsDefinitions

TermTerm DegreeDegree

2x2x33 335x5x66 66xx 11xyxy 22

Lecture 28Lecture 28 7575

Degree of a Degree of a polynomial:polynomial:Examples:Examples:

DefinitionsDefinitions

Lecture 28Lecture 28 7676

Degree of a Degree of a polynomial:polynomial:

Greatest degree of any term of Greatest degree of any term of the polynomialthe polynomial

Examples:Examples:

DefinitionsDefinitions

Lecture 28Lecture 28 7777

Degree of a Degree of a polynomial:polynomial:

Greatest degree of any term of Greatest degree of any term of the polynomialthe polynomial

Examples:Examples:

DefinitionsDefinitions

PolynomialPolynomial DegreeDegree

2x2x33+7x+9+7x+9

Lecture 28Lecture 28 7878

Degree of a Degree of a polynomial:polynomial:

Greatest degree of any term of Greatest degree of any term of the polynomialthe polynomial

Examples:Examples:

DefinitionsDefinitions

PolynomialPolynomial DegreeDegree

2x2x33+7x+9+7x+9 33

Lecture 28Lecture 28 7979

Degree of a Degree of a polynomial:polynomial:

Greatest degree of any term of Greatest degree of any term of the polynomialthe polynomial

Examples:Examples:

DefinitionsDefinitions

PolynomialPolynomial DegreeDegree

2x2x33+7x+9+7x+9 335x5x6 6 –2x–2x44+9x -7+9x -7

Lecture 28Lecture 28 8080

Degree of a Degree of a polynomial:polynomial:

Greatest degree of any term of Greatest degree of any term of the polynomialthe polynomial

Examples:Examples:

DefinitionsDefinitions

PolynomialPolynomial DegreeDegree

2x2x33+7x+9+7x+9 335x5x6 6 –2x–2x44+9x -7+9x -7 66

Lecture 28Lecture 28 8181

Degree of a Degree of a polynomial:polynomial:

Greatest degree of any term of Greatest degree of any term of the polynomialthe polynomial

Examples:Examples:

DefinitionsDefinitions

PolynomialPolynomial DegreeDegree

2x2x33+7x+9+7x+9 335x5x6 6 –2x–2x44+9x -7+9x -7 66x + y + 6x + y + 6

Lecture 28Lecture 28 8282

Degree of a Degree of a polynomial:polynomial:

Greatest degree of any term of Greatest degree of any term of the polynomialthe polynomial

Examples:Examples:

DefinitionsDefinitions

PolynomialPolynomial DegreeDegree

2x2x33+7x+9+7x+9 335x5x6 6 –2x–2x44+9x -7+9x -7 66x + y + 6x + y + 6 11

Lecture 28Lecture 28 8383

Degree of a Degree of a polynomial:polynomial:

Greatest degree of any term of Greatest degree of any term of the polynomialthe polynomial

Examples:Examples:

DefinitionsDefinitions

PolynomialPolynomial DegreeDegree

2x2x33+7x+9+7x+9 335x5x6 6 –2x–2x44+9x -7+9x -7 66x + y + 6x + y + 6 11xy + 5x - 9yxy + 5x - 9y

Lecture 28Lecture 28 8484

Degree of a Degree of a polynomial:polynomial:

Greatest degree of any term of Greatest degree of any term of the polynomialthe polynomial

Examples:Examples:

DefinitionsDefinitions

PolynomialPolynomial DegreeDegree

2x2x33+7x+9+7x+9 335x5x6 6 –2x–2x44+9x -7+9x -7 66x + y + 6x + y + 6 11xy + 5x - 9yxy + 5x - 9y 22

Lecture 28Lecture 28 8585

What is a binomial?What is a binomial?

Lecture 28Lecture 28 8686

What is a binomial?What is a binomial?

A polynomial with exactly two terms.A polynomial with exactly two terms.

Lecture 28Lecture 28 8787

What is a binomial?What is a binomial?

A polynomial with exactly two terms.A polynomial with exactly two terms.

Examples: x + y, 3 + x, 4xExamples: x + y, 3 + x, 4x22 + 9 + 9

Lecture 28Lecture 28 8888

To Multiply Binomials, use FOIL:To Multiply Binomials, use FOIL:

(ax+b)(cx+d)(ax+b)(cx+d)

Lecture 28Lecture 28 8989

To Multiply Binomials, use FOIL:To Multiply Binomials, use FOIL:

acxacx2 2 ++

(ax+b)(cx+d)(ax+b)(cx+d)FF

Lecture 28Lecture 28 9090

To Multiply Binomials, use FOIL:To Multiply Binomials, use FOIL:

acxacx2 2 + + adxadx

(ax+b)(cx+d)(ax+b)(cx+d)F O

O

F

Lecture 28Lecture 28 9191

To Multiply Binomials, use FOIL:To Multiply Binomials, use FOIL:

acxacx2 2 + + adx adx + + bcxbcx

(ax+b)(cx+d)(ax+b)(cx+d)F O I

I

O

F

Lecture 28Lecture 28 9292

To Multiply Binomials, use FOIL:To Multiply Binomials, use FOIL:

acxacx2 2 + + adx adx + + bcx bcx ++ bdbd

(ax+b)(cx+d)(ax+b)(cx+d)L F O I L

I

O

F

Lecture 28Lecture 28 9393

To Multiply Binomials, use FOIL:To Multiply Binomials, use FOIL:

acxacx2 2 + + adx adx + + bcx bcx ++ bdbd

(ax+b)(cx+d)(ax+b)(cx+d)

acxacx2 2 + (ad+bc)+ (ad+bc)xx ++ bdbd

L F O I LI

O

F

Lecture 28Lecture 28 9494

Multiply: Multiply: (x+5)(x+3)(x+5)(x+3)

(x+5)(x+3)(x+5)(x+3)

Lecture 28Lecture 28 9595

Multiply:Multiply:

xx22

(x+5)(x+3)(x+5)(x+3)F

Lecture 28Lecture 28 9696

Multiply:Multiply:

xx2 2 + + 3x3x

(x+5)(x+3)(x+5)(x+3)F O

Lecture 28Lecture 28 9797

Multiply:Multiply:

xx2 2 + + 3x 3x + + 5x5x

(x+5)(x+3)(x+5)(x+3)F O I

Lecture 28Lecture 28 9898

Multiply:Multiply:

xx2 2 + + 3x 3x + + 5x 5x ++ 1515

(x+5)(x+3)(x+5)(x+3)F O I L

Lecture 28Lecture 28 9999

Multiply:Multiply:

xx2 2 + + 3x 3x + + 5x 5x ++ 1515

(x+5)(x+3)(x+5)(x+3)

xx2 2 + + 8x8x ++ 1515

F O I L

Lecture 28Lecture 28 100100

Multiply: Multiply: (x-6)(x+2)(x-6)(x+2)

Lecture 28Lecture 28 101101

Multiply: Multiply: (x-6)(x+2)(x-6)(x+2)

xx22

(x-6)(x+2)(x-6)(x+2)

Lecture 28Lecture 28 102102

Multiply: Multiply: (x-6)(x+2)(x-6)(x+2)

xx2 2 + + 2x2x

(x-6)(x+2)(x-6)(x+2)

Lecture 28Lecture 28 103103

Multiply: Multiply: (x-6)(x+2)(x-6)(x+2)

xx2 2 + + 2x 2x - - 6x6x

(x-6)(x+2)(x-6)(x+2)

Lecture 28Lecture 28 104104

Multiply: Multiply: (x-6)(x+2)(x-6)(x+2)

xx2 2 + + 2x 2x - - 6x 6x -- 1212

(x-6)(x+2)(x-6)(x+2)

Lecture 28Lecture 28 105105

Multiply: Multiply: (x-6)(x+2)(x-6)(x+2)

xx2 2 + + 2x 2x - - 6x 6x -- 1212

(x-6)(x+2)(x-6)(x+2)

xx2 2 - - 4x4x -- 1212

Lecture 28Lecture 28 106106

Multiply: Multiply: (x-7)(x-5)(x-7)(x-5)

Lecture 28Lecture 28 107107

Multiply: Multiply: (x-7)(x-5)(x-7)(x-5)

xx22

(x-7)(x-5)(x-7)(x-5)

Lecture 28Lecture 28 108108

Multiply: Multiply: (x-7)(x-5)(x-7)(x-5)

xx2 2 - - 5x5x

(x-7)(x-5)(x-7)(x-5)

Lecture 28Lecture 28 109109

Multiply: Multiply: (x-7)(x-5)(x-7)(x-5)

xx2 2 - - 5x 5x - - 7x7x

(x-7)(x-5)(x-7)(x-5)

Lecture 28Lecture 28 110110

Multiply: Multiply: (x-7)(x-5)(x-7)(x-5)

xx2 2 - - 5x 5x - - 7x 7x ++ 3535

(x-7)(x-5)(x-7)(x-5)

Lecture 28Lecture 28 111111

Multiply: Multiply: (x-7)(x-5)(x-7)(x-5)

xx2 2 - - 5x 5x - - 7x 7x ++ 3535

(x-7)(x-5)(x-7)(x-5)

xx2 2 - - 12x12x ++ 3535

Lecture 28Lecture 28 112112

Multiply:Multiply:

6x6x22

(2x+5)(3x-8)(2x+5)(3x-8)

Lecture 28Lecture 28 113113

Multiply:Multiply:

6x6x2 2 - - 16x16x

(2x+5)(3x-8)(2x+5)(3x-8)

Lecture 28Lecture 28 114114

Multiply:Multiply:

6x6x2 2 - - 16x 16x + + 15x15x

(2x+5)(3x-8)(2x+5)(3x-8)

Lecture 28Lecture 28 115115

Multiply:Multiply:

6x6x2 2 - - 16x 16x + + 15x 15x -- 4040

(2x+5)(3x-8)(2x+5)(3x-8)

Lecture 28Lecture 28 116116

Multiply:Multiply:

6x6x2 2 - - 16x 16x + + 15x 15x -- 4040

(2x+5)(3x-8)(2x+5)(3x-8)

6x6x2 2 - - xx -- 4040

Lecture 28Lecture 28 117117

Multiply:Multiply:

6x6x2 2 - - 16x 16x + + 15x 15x -- 4040

(2x+5)(3x-8)(2x+5)(3x-8)

6x6x2 2 - - xx -- 4040

Lecture 28Lecture 28 118118

Multiply: (3x+4)Multiply: (3x+4)22

9x9x2 2

(3x+4)(3x+4)(3x+4)(3x+4)

Lecture 28Lecture 28 119119

Multiply: (3x+4)Multiply: (3x+4)22

9x9x2 2 + + 12x12x

(3x+4)(3x+4)(3x+4)(3x+4)

Lecture 28Lecture 28 120120

Multiply: (3x+4)Multiply: (3x+4)22

9x9x2 2 + + 12x 12x + + 12x12x

(3x+4)(3x+4)(3x+4)(3x+4)

Lecture 28Lecture 28 121121

Multiply: (3x+4)Multiply: (3x+4)22

9x9x2 2 + + 12x 12x + + 12x 12x ++ 1616

(3x+4)(3x+4)(3x+4)(3x+4)

Lecture 28Lecture 28 122122

Multiply: (3x+4)Multiply: (3x+4)22

9x9x2 2 + + 12x 12x + + 12x 12x ++ 1616

(3x+4)(3x+4)(3x+4)(3x+4)

9x9x2 2 + 24+ 24xx ++ 1616

Lecture 28Lecture 28 123123

Lecture 28Lecture 28 124124

Lecture 28Lecture 28 125125