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Math 30 > U1 A1- Polynomial Functions And Operations > Flashcards

U1 A1- Polynomial Functions And Operations Flashcards

1
Q

Do you factor the simplified expression once you add/subtract polynomial functions

A

No

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2
Q

In f(x)-g(x) what do you have to remember

A

Put ALL of g(x) into brackets and apply the negative to the whole expression

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3
Q

What is the order of operations for -x^2

A

1) square first

2) then apply the negative sign

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4
Q

What does it mean when a function is defined

A

It has an equation

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5
Q

When functions aren’t defined how do you find the sum or difference

A

Add/subtract the y values (outputs) at a given x value (input)

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6
Q

Given:

F(-3)=5

g(-3)=8

What is f(-3) + g(-3)

A

13

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7
Q

The domain of the combined sum or difference function may be ______ than the _____ domain

The range of the combined function can be determined from the _____

A

Different

Individual

Graph

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8
Q

F(x) domain: (-infinite,3)

g(x) domain: (-5,infinite)

What is the domain of (f+g)(x)

A

(-5,3)

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9
Q

The domain and range of a graph is always ____ unless there are restrictions

A

Infinite

x is an element of real numbers

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10
Q

How do you know if domain and range aren’t infinite when just given equation

A

If there are restrictions in the equation

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11
Q

What are 2 examples of restrictions on an equation

A
  • x under a square root sign

- dividing something by x

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12
Q

What is the constant of a quadratic function (in terms of graphing)

A

y-intercept

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13
Q

The domain of the product or quotient function may be ______ than the _____

A

Different

Individual

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14
Q

The range of a combined function (multiplication or division) can be determined from the ____

A

Graph

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15
Q

How do you draw a new graph when you combine two functions

A

1) make a table and state the y-value of f(x) and g(x) at x values that you pick
2) combine the two y-values however you’re supposed to (add/sub/mul/div) And those are your new y-values

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16
Q

When you are determined a simplified expression for multiplication or division of polynomials should you factor after

A

No

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17
Q

How to determine the degree of a function multiplication statement without foiling it out

A

Count the x’s

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18
Q

(Linear function )(linear function)= _________

and why

A

Quadratic

(x)(x)=x^2

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19
Q

What is always the domain of a quadratic unless there are restrictions

A

Infinite

x is an element of real numbers

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20
Q

When is the only time you can add/subtract radicals

A

When the radicand are like terms

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21
Q

What is a radicand

A

Stuff under root sign

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22
Q

Before you state NPV’s what do you always have to do

A

Factor the expression

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23
Q

A discontinuity is a _____ in the graph

A

Graph

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24
Q

Vertical asymptote- a line that the graph _____ but never touches

A

Approaches

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25
Q

A vertical asymptote occurs when a ration expression cannot be ______ further

A

Simplified

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26
Q

Identify the vertical asymptote in

y=(x-4)/(x-1)

A

Restriction: x can’t equal (1)

Since ORIGINAL EXPRESSION can’t simplify further,
x= (-1) is vertical asymptote

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27
Q

Point of discontinuity- a single ___ that doesn’t exist on the graph

A

Point

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28
Q

A point of discontinuity occurs when a rational expression ___ be simplified further

A

Can

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29
Q

Identify the point of discontinuity in the expression

y= [(x-1)(x+2)]/(x-1)

A

Restriction: x can’t equal (1)

Expression can be simplified further to (x+2) so x=1 is a point of discontinuity

Then sub the x value into the expression to get the y-value

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30
Q

The y-value of a point of discontinuity can be found by _______ the restricted x-value into the _____ expression

A

Substituting

Simplified

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31
Q

What is quotient to anything divided by 0

And what does this mean in terms of functions

A

Undefined

Know that this is a point of discontinuity

(look at the x-value and plug it in to find the y value of the hole)

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32
Q

(Quadratic function)/(linear function)=_______

and why

A

Linear function

X^2)/(x)=(x

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33
Q

How do you solve 3(fg)(2)

A

1) evaluate the product of (f)(g) when x is 2

2) multiply the whole thing by 3

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34
Q

When adding/subtracting polynomials fractions you need ______

A

LCD’s

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35
Q

Can you cancel binomials in division if they’re not the EXACT same expression

A

No

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36
Q

When multiplying/dividing functions a NPV will result in a ______ or _____

A

Vertical asymptote

Point of discontinuity

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37
Q

To draw the graph of functions being multiplied/divided you multiply the two ____ values at the same ____ value

A

Y

X

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38
Q

Composite functions- ____ within functions

A

Functions

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39
Q

How do you write (f dot g)(x) as a composite function

A

f[g(x)]

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40
Q

To find the equation for composite functions, ______ the inside function for x in the ____ function

A

Substitute

Outside

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41
Q

To evaluate composite functions, evaluate the ____ function at the given ___ value then take this output and evaluate the _____ function at the ____ x value

A

Inside

X

New

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42
Q

How to find domain of composite functions

A

Combine the domains of each function

Identify any restrictions and state domain (as we always do)

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43
Q

Composite functions- if the inner function isn’t defined, neither is the _____ function

A

Composite

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44
Q

What does a dot with no colour mean

A

A functions OF a function

45
Q

How to determine f[g(4)] using a graph

A

1) Find y value of g(x) graph at (4)

2) find y value of f)x) graph at whatever the new x value is

46
Q

Polynomial functions are characterized by their ____, _____, and constant term

A

Degree

Leading coefficient

47
Q

Graphs- odd degree polynomials have ____ arms

A

Opposite

48
Q

Odd degree polynomials- if the leading coefficient is GREATER than 0, the graph will ____ from left to right

A

Rise

49
Q

Odd degree polynomials- if the leading coefficient is LESS than 0, the graph will ____ from left to right

A

Fall

50
Q

Graphs-even degree polynomials have arms going the ____ direction

A

Same

51
Q

Are negatives whole numbers

A

Yes

52
Q

Even degree polynomials- if the leading coefficient degree is GREATER than 0, the graph will open ____

A

Up

53
Q

Even degree polynomials- if the leading coefficient degree is LESS than 0, the graph will open ____

A

Down

54
Q

What does the constant term represent in terms of graphing

A

Y-intercept

55
Q

All polynomial functions exhibit _______

Abs what does this mean

A

Continuity

No breaks in the graph, so no restrictions on the domain

56
Q

Polynomial functions- all tents MUST have ____ number coefficients

A

Whole

57
Q

Polynomial functions- the leading coefficient must be a ___ number

A

Real

58
Q

What is a real number

A

Can be defined

59
Q

How do you draw a graph for a function given the characteristics of the function

A

Look at degree, LC, and constant

Draw a graph based off of the identified characteristic in each of those

60
Q

When drawing a graph if the constant term is negative what do you have to remember

A

The y-intercept has to be negative

61
Q

Odd degree functions- arms go in _____ directions

A

Opposite

62
Q

Even degree functions- arms go in ____ direction

A

Equal

63
Q

Max # of turning points of a graph is less than the ______

A

Degree

64
Q

What is the dividend

A

that is being divided

65
Q

What is the divisor

A

that another # is being divided by

66
Q

What is the non- simplified formula for division of polynomials

A

(dividend/divisor)=Q +(remainder/

divisor)

67
Q

What is the simplified formula for division of polynomials

A

dividend= divisor(Q) + remainder

68
Q

WHAT IS THE MOST IMPORTANT THING WITH LONG DIVISON OF POLYNOMIALS

(IF YOU DONT DO THIS YOUR ANSWER WILL BE WRONG)

A

Any powers of the variable that are missing must be included with a 0 coefficient

Ex) 3x^3 + 0x^2 + x +9

69
Q

What is the remainder Therom

A

When P(x) is divided by a binomial, the remainder can be determined by evaluating P(a), where a is the root of the divisor

70
Q

What is the formula for remainder therom

And what do the variables represent

A

P(a)= R

a= root of the divisor

R=remainder

71
Q

When question gives you dividend and divider and asks for remainder, should you use long divison

(And why)

A

NO

(Use remainder therom

72
Q

Steps for using remainder theorem to find the remainder

A

1) find the root of the divisor
(Let equation equal 0 and solve)

2) plug that value (a) into dividend to solve for remainder

73
Q

Factoring polynomials: if a number is a factor of another number, it can perfectly divide into the second number without a ____

A

Remainder

74
Q

The factor theorem is an _____ of the remainder theorem

A

Extension

75
Q

The factor theorem: when a polynomial is divided by a binomial such that the _____ is 0, then the binomial ____ a factor of the polynomial

A

Remainder

Is

76
Q

Factoring polynomials: the root of the divisor is a ___ or ___ ____ of the polynomial

(Means it’s both)

A

Zero

X-intercept

77
Q

Factoring polynomials: what does a 0 output mean

A

You have a perfect root, so a perfect factor

78
Q

Steps for factoring polynomials degree 3 or higher

A

1) list all the factors of the constant term (pos and neg)

2) evaluate the expression at each of the factors until one of them =0
(Sub number in for all the x values)

3) find the factor from that since
x= whatever number was correct

4) divide expression by that factor

5)
- if quotient is degree two use decomp to find the other two factors

  • if quotient is degree three or higher repeat the same process again (keeping in mind that your final answer will have that first factor you divided by in it)
79
Q

How to find the roots, given the factors

A

let each factor equal 0 and solve for x

80
Q

How are the quadrants labelled on a graph

A

Top right=1

Top left=2

81
Q

What is the min and max number of zeros a degree 5 polynomial can have

(And why)

A

1-5

  • odd degree means it has opposite arms (so has to cross x axis at least once)
  • 5 since it’s degree five
82
Q

What is the min and max number of zeros a degree 4 polynomial can have

(And why)

A
  • even degree means it’s arms go in the same direction (so doesn’t have to cross x axis even once)
  • 4 since it’s a degree 4
83
Q

Multiplicity refers to how many times a particular ____ of a function ____

A

Zero

Repeats

84
Q

WHAT 4 THINGS DO YOU HABE TO INDENTIFY IN AN EQUATION IN ORDER TO DRAW THE CORRESPONDING GRAPH

A
  • multiplicity
  • degree
  • end behaviour
  • constant
85
Q

What is the end behaviour of a graph and how do you determine it

(Both odd and even degree polynomials)

A

If the graph falls or rises

Odd:

Negative lead coefficient= falls
Positive lead coefficient= rises

Even:

Negative lead coefficient= opens down
Postive lead coefficient=rises

86
Q

What does a multiplicity of 2 mean

A

The factor repeats two times

87
Q

Multiplicity: what is the behaviour if there’s a multiplicity of 1

A

Crosses x axis

At whatever x equals

88
Q

Multiplicity: what is the behaviour if there’s a multiplicity an even #

A

Touches x axis

At whatever x equals

89
Q

Multiplicity: what is the behaviour if there’s a multiplicity an odd #

A

Bends at x axis

At whatever x equals

90
Q

Multiplicity: after you determined the behaviour, how do you know where the behaviour happens

A

Find the root of the factor

91
Q

Multiplicity: how many different behaviours are there

A

3

92
Q

How to determine a transformation equation from a point on the original graph

A

Pick a point on the original graph

1) determine stretch
2) apply transformation to point
3) determine reflection
4) apply transformation to point
5) determine translation
6) apply transformation to that point

93
Q

WHAT CAN’T YOU DO WHEN DETERMINING A TRANSFORMATION FROM AN ORGINAL AND TRANSFORMED GRAPH

A

Pick a point on original and pick corresponding point on transformed and determine equation from that

94
Q

How to solve:

For what value of b will
P(x)= 4x^3 -3x^2 + bx +6
Have the same remainder when it is divided by both (x-1) and (x+3)

A

1) Sub each root in the equation and simply

3) P(1)=P(-3) and solve for b

95
Q

Steps for solving by substitution

A

1) simplify two equations (there should be at least one x and y in each)
2) manipulate one equation (doesn’t matter which), so that one variable is in terms of the other
3) solve that equation (now you have the value of one variable)
4) sub the value you just found into the other simplified equation (to determine the value of the other variable

96
Q

If h(x)= f[g(x)] determine f(x) and g(x) when:

h(x)= 2x^2 -1

A

1) g(x) has to have an x in it in Oder to be a function so let it equal x^2
2) work backwards from there (you can see that x^2 is being subbed into 2x-1) so that is the function of f(x)

97
Q

How to solve:

For what value of c will
P(x)= 2x^2+ cx
Have the same remainder when it’s divided by x-2 and x+1

A

1) find the two roots
2) make each expression equal each other and sub in the roots p(-1)=p(2)
3) simplify and solve for c

98
Q

How to do this:

When 3x^2 + 6x -10 is divided by (x+k), the remainder is 14. What is the value(s) of k

A

1) start as usual with P(a)=R

2) make equation equal 0 after you’ve simplified and factor to find values of k

99
Q

How to do this:

When the polynomial
mx^3 - 3x + nx +2 is divided by (x-2) the remainder is 1. When it’s divided by (x+3) the remainder is 4.

What are the values of m and n

A

1) sub the roots in and simplify the equation as much as you can
2) add the equations to find the value of one variable
3) sub that value onto one of the simplified equations to find the value of the other variable

100
Q

How to do this:

If h(x)=f[g(x] 
and h(x)=(2x-5)^2 and f(x)= x^2 what is g(x)

And what’s the answer

A

Inside term is always going to be the inside function so g(x)=2x-5

And you can clearly see this makes sense if f(x)=x^2

101
Q

When you’re dividing by two fractions how do you set it up in a way that you can easily solve

A

Multiply the top fraction by the reciprocal of the bottom fraction

102
Q

How to find the new domain and range of a function after you add/subtract/divide/multiply

A

Graph it

103
Q

What are 3things that make a polynomial function NOT a polynomial function

A
  • x under the root sign
  • fraction exponent
  • negative exponent
104
Q

If a number is divided just by x is it a polynomial function

A

No (it has a discontinuity)

105
Q

If part of a polynomial function is -2x^2 with the -2 under the root sign is it a polynomial

A

No (anything squared is positive and positive times negative is negative, which you can have under a root sign)

106
Q

What is a rational function

A

Both numerator and denominator are polynomials

107
Q

What is a radical function

A

Has a square root with x in the radicand

108
Q

When finding multiplicties if there’s an x by itself what do you have to do

A

Include a multiplicity with that as well

Math 30 (6 decks)

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