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


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


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


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



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



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



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



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



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



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) =) =



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


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


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..



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

tt hh22 256256

55 400400

88 256256

1010 00



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..


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

100

400

200

300

2 4 6 Time

tt hh22 25625655 40040088 256256

1010 00

8 10

Height


Term:Term:DefinitionsDefinitions


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



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





Like TermsLike Terms





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.





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



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




-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


Simplify:Simplify:

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


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


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


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


Simplify:Simplify:

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


Simplify:Simplify:

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

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


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


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


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




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

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




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

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




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




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




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


Polynomial:Polynomial:DefinitionsDefinitions


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





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





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

xx



Polynomials:Polynomials: Nonpolynomials:Nonpolynomials:


2

3 12 3 5

xx x


Polynomials:Polynomials: Nonpolynomials:Nonpolynomials:


2

3 12 3 5

xx x

3 17x


Polynomials:Polynomials: Nonpolynomials:Nonpolynomials:DefinitionsDefinitions

2

3 12 3 5

xx x

2x

3 17x



2

3 12 3 5

xx x

2x23

3 17x



2

3 12 3 5

xx x

2x

3 42 3xx yy

23

3 17x



2

3 12 3 5

xx x

2x

3 42 3xx yy

23

2 23 2x xy y

3 17x



2

3 12 3 5

xx x

2x

3 42 3xx yy

1x

23

2 23 2x xy y

3 17x



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


Monomial:Monomial:Examples:Examples:

Binomial:Binomial:Examples:Examples:

Trinomial:Trinomial:Examples:Examples:



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

Examples:Examples:






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:







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:







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



Degree of a term:Degree of a term:

Examples:Examples:



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:




Examples:Examples:


TermTerm DegreeDegree

2x2x33



Examples:Examples:



2x2x33 33



Examples:Examples:



2x2x33 335x5x66



Examples:Examples:



2x2x33 335x5x66 66



Examples:Examples:



2x2x33 335x5x66 66xx



Examples:Examples:



2x2x33 335x5x66 66xx 11



Examples:Examples:



2x2x33 335x5x66 66xx 11xyxy



Examples:Examples:



2x2x33 335x5x66 66xx 11xyxy 22


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



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:





Examples:Examples:


PolynomialPolynomial DegreeDegree

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




Examples:Examples:



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




Examples:Examples:



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




Examples:Examples:



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




Examples:Examples:



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




Examples:Examples:



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




Examples:Examples:



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




Examples:Examples:



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


What is a binomial?What is a binomial?



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



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


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

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



acxacx2 2 ++

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



acxacx2 2 + + adxadx

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

O

F



acxacx2 2 + + adx adx + + bcxbcx

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

I

O

F



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

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

I

O

F



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


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

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


Multiply:Multiply:

xx22

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


Multiply:Multiply:

xx2 2 + + 3x3x

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


Multiply:Multiply:

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

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


Multiply:Multiply:

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

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


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


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



xx22

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



xx2 2 + + 2x2x

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



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

(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 + + 2x 2x - - 6x 6x -- 1212

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

xx2 2 - - 4x4x -- 1212


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



xx22

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



xx2 2 - - 5x5x

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



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

(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 - - 5x 5x - - 7x 7x ++ 3535

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

xx2 2 - - 12x12x ++ 3535


Multiply:Multiply:

6x6x22

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


Multiply:Multiply:

6x6x2 2 - - 16x16x

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


Multiply:Multiply:

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

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


Multiply:Multiply:

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

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


Multiply:Multiply:

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

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

6x6x2 2 - - xx -- 4040


Multiply:Multiply:

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

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

6x6x2 2 - - xx -- 4040


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

9x9x2 2

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



9x9x2 2 + + 12x12x

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



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

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



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

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



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

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

9x9x2 2 + 24+ 24xx ++ 1616




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

Documents