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Lecture 283 WT shoots an arrow straight up with an initial velocity of 160 feet per second. The height (in feet) of the arrow is given by the equation: h = -16t t t is the number of seconds the arrow is in the air. Find the height of the arrow. t h=-16t t
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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