Question

im a little confused about the zero product property
suppose that you are asked to solve
x^2 + 5x + 6 = 0
so we factor
(x+3)(x+2) = 0 , and the zero product theorem says that x = -3 or x = -2. but i dont see why this is a solution to the original polynomial. and why isnt it x = -3 and x = -2, why or?

Answer 1

so i am asking two different questions

Answer 2

because 'and' means that x is BOTH values AT THE SAME TIME, which is impossible

the 'or' means that x is either one value or the other (but not at the same time)

Answer 3

not both at the same time*

Answer 4

ok, and why are these the solutions to the original polynomial ?

Answer 5

if you replace x with either -3 OR -4, and evaluate/simplify, you'll get 0

Answer 6

But the "or" doesn't mean that they aren't both solutions, right? It's just that x can't be both values at the same time, right?

Answer 7

why does factoring the polynomial and saying x = a or x = b, how come that solves the original equation

Answer 8

the "or" means that either one is a solution, but you use it over "and" to avoid saying x is both values at the same time

Answer 9

so the "or" is implying that both are the solutions (just not at the same time)

Answer 10

why is x = a or x = b solutions
to x^2 - (a+b)x + ab = 0 ?

Answer 11

the poly factors into the product of to binoms

Answer 12

*two

Answer 13

but the funny thing is, if x = 2 or x = 3, its possible that x is not equal to 2 , but x = 3.

Answer 14

funny thing?

Answer 15

x^2 - (a+b)x + ab = 0

a^2 - (a+b)a + ab = 0 ... Replace all 'x' terms with 'a' (since x = a is one solution)

a^2 - (a^2 + ab) + ab = 0

a^2 - a^2 - ab + ab = 0

0 = 0

So this shows that x = a is a solution to x^2 - (a+b)x + ab = 0

Answer 16

well if x = 3, then x can't equal 2 (x can only equal one thing)

Answer 17

x is any value from -inf to inf; it just so happens that when x=-2, the function equals zero; and then when x=3, the function equals zero

Answer 18

yes, but why in general does factoring work for the original polynomial

Answer 19

but x can certainly equal 2

Answer 20

yes but it is 'or' so x = -2 OR x = -3, so its possible that x is not equal to -3 ...
ok i think im missing some theorem here .
is this a factor theorem,

Answer 21

weve been multiplying things since what, the 3rd grade?

x+3
x+2
-----
x^2+3x
+2x+6
-----------
x^2+5x+6

Answer 22

if x is not equal to -2, then x is equal to -3. if x is not equal to -2, then x is equal to -3. x cannot be both equal to -2 and -3.
ok but im missing something, how do we know this solves the original polynomial
without actually substituting (long way)

Answer 23

the original equation is
x^2 + 5x + 6 = 0,
so im asking, why does solving the factored form of the polynomial
solve the original polynomial equation

Answer 24

the zero factor thrm ... also heard it called the zero product rule; is what happens when we multiply a term by zero, we get a zero back

Answer 25

ok, you get a zero back. but how does that show the original polynomial (expanded) is a solution

Answer 26

when does ab = 0? when either a or b (or both) are zero

0*b = 0
a*0 = 0
0*0 = 0

we learned this form zero multiplication tables

Answer 27

but these are different expressions. one is the polynomial x^2 + 5x + 6,
the other is (x+2)(x+3), they are very different

Answer 28

when does

x^2 + 5x + 6 = 0 ??

is the same as saying when does:

(x+2)(x+3) = 0 ; well, when x+2 = 0 or when x+3 = 0

Answer 29

why does the zeroes of (x+2)(x+3) make x^2+5x+6 also zero?

Answer 30

why do you insist that they are different?

Answer 31

becasue when you multiply out the products you get the poly .... as srawned out in a previous post

Answer 32

but , i'm not comfortable with your logic
you prove that x^2+5x+6 = 0 when x +2 = 0 or x + 3 = 0.
so that means x could equal to 50 or -3 (since it is an 'or' statement )

Answer 33

the truth value of the statement ' x = -3 or x = -2 ' is equivalent to the truth value of ' x = 50 or x = -2' . since they are both true when x = -2

Answer 34

would you rather do this with math instead of making up nonmath things to try to confound the issue?

Answer 35

an 'or' statement means that only one of the parts has to be true.
im not making up non -math, i dont like your tone

Answer 36

what does (x+2)(x+3) equal?

Answer 37

this is basic logic, you should study it

Answer 38

your not applying logic either

Answer 39

yes i know it is equal to x^2 + 5x + 6. im asking about the 'or' statement

Answer 40

yes i am, 'p or q' is true when p is true.

Answer 41

'p or q' is true when p is true and q is false

Answer 42

the or statement says that when x= one or the other, then we get a zero; you are trying to say that x can be more than one value at a time

Answer 43

so ' x = -3 or x = 50 ' is true when x = -3 , so is
' x = -3 or x = -2 '

Answer 44

no im saying one is enough , one solution

Answer 45

are these steps reversible, maybe thats what im looking for

Answer 46

we want two solutions, but the 'or' only guarantees one solution is true

Answer 47

you might be confusing the english use of or as an exclusive or instead of that other one

Answer 48

x is either one or the other in order for this to be zero

Answer 49

' p or q ' is true when
p is true, q is false
p is false, q is true
p is true, q is true

Answer 50

we cannot infer from
' john is 8 years old or john live in california' that both are true. we only know at least ONE is true

Answer 51

so saying
' x = -2 or x = -3' only tells me ONE of these are the solutions.

Answer 52

but i dont know which one ( we can never know )

Answer 53

strictly from logic :)

Answer 54

and you also seem to be stretching the logic; 8 years and california are not both in the same domain

John is 8 years old or John is 32 years old

Answer 55

so again we cannot infer from 'p or q' , you cannot infer p is true or that q is true.

Answer 56

i wanted to use different propositions

Answer 57

'john is 8 years old or mike is 24 ' , you cannot infer that john is 8 , and you cannot infer that mike is 24

Answer 58

but the original setup doesnt use different propositions; you have to construct the analogy to match the original or else its useless

Answer 59

we dont have 2 different variables; no mike and john; just john

Answer 60

well so far we have an 'or' statement. so we cannot infer which one is the solution, do you know what i mean. all we can say is that one of them is the solution

Answer 61

ok, 'john is 8 years old or john is 24 '

Answer 62

your misuse of logic aside, the original statement is sound :)

Answer 63

we dont know which of those is true, all we know is that if the 'or' statement is true, one of them is true

Answer 64

Let say John is a function defined as when John is 8 or when John is 32; he lives at home, otherwise he lives someplace else that isnt home

when does john live at home?

Answer 65

hmm

Answer 66

what does living at home have to do with age

Answer 67

what does that matter, its a rule that defines a function

Answer 68

im not sure

Answer 69

as we journey along the poly curve, there are only specific values of x that we come across that make the function zero

Answer 70

but all im saying is, when you have an 'or' statement, you cannot know which of them is true . p or q is true , we dont know if p is true or q is true

Answer 71

there are only certain times in johns live that he lives at home

Answer 72

, all we can say is at least one of p or q is true (possibly both, depending on the problem)

Answer 73

If you want to do this from a strict logical standpoint, then you might do something like this


There are some numbers that satisfy x^2 + 5x + 6 = 0


There are some numbers that satisfy (x+3)(x+2) = 0


There are some numbers that satisfy x+3 = 0 or x+2 = 0


There are some numbers that satisfy x = -3 or x = -2


Keep in mind that (p v q) <==> (q v p), so we can easily say "There are some numbers that satisfy x = -2 or x = -3"


BUT

You cannot say (p v q) <==> (p v r), which is why you can't go from


"There are some numbers that satisfy x = -3 or x = -2"

to

"There are some numbers that satisfy x = -3 or x = -50"

Answer 74

but im saying, suppose we label p: ' x = -2' q: ' x = -3' ,
and that we say 'p or q' is true.

Answer 75

we don't know which one is true , whether x is equal to -2 or whether x = -3 ,

Answer 76

if (x=a or x=b), then (f(x)=0) is the statement to focus
on; this gives the proper form for p and q

Answer 77

both are true because both are part of the solution set

Answer 78

hmm, but an 'or' statement only guarantees one of them is true. and we dont know which

Answer 79

if you solve x^2 + 5x + 6 = 0, you get the solution set {-2, -3}

or you can start with the idea that the solution set is {-2, -3} to get k(x^2 + 5x + 6) = 0 for some constant k

Answer 80

there are 2 or statments that are used in english

Answer 81

if i say it is true that ' john lives in germany or john is 28' we dont know which one is true. john could live in australia and be 28 , and it still be true

Answer 82

or john could be 12 and live in germany, and it is still true

Answer 83

but i might be stretching the xor assessment :)

Answer 84

when john is 12, does he live at home? which is what the function is asking for; when does john live at home

Answer 85

well im not so concerned with the exclusive or, since we said earlier that x can't be both -2 and -3 at the same time anyway

Answer 86

' mr. smith likes apple pie or mr. smith likes spinach' . we dont know which one

Answer 87

all we can say is that mr. smith likes at least one of {apple pie, spinach }

Answer 88

possibly both

Answer 89

if (x=a or x=b), then (f(x)=0)

if (f(x) !=0 ) then (x!=a and x!=b) has the same truth value since its a contrapositive

Answer 90

the way you are using it, still has a problem.

Answer 91

if (x=a or x=b), then (f(x)=0)

has the same truth value as

if (f(x)=0), then (x=a or x=b) which means that we can construct an iff statment, biconditional

Answer 92

' mr. smith likes apple pie or mr. smith likes spinach' . we dont know which one

Answer 93

your misusing the phraseology again :)

Answer 94

' x = -3 or x = -2' , we dont know which one

Answer 95

if (smith eats apples or smith eats spinach); THEN __________

if (smith eats apples or smith eats spinach); then (smith is eating supper)

Answer 96

propositions are the ors ands right?

Answer 97

we want to deduce that x = { -3, -2} , we cannot deduce that from x = -2 or x = -3

Answer 98

Another way to look at it


"{m,n} is the solution set to x^2 + 5x + 6 = 0" ---> True (all quadratics have 2 solutions)


"{m,n} is the solution set to (x+3)(x+2) = 0" ---> True ((x+3)(x+2) is equivalent to x^2 + 5x + 6)


"{m,n} is the solution set to x+3=0 or x+2 = 0" ---> True (zero product property)


"{m,n} is the solution set to x = -3 or x = -2" ---> True (Additive property of equality)


If you work backwards (then forward again), you can say


"{-3, -2} is the solution set to x^2 + 5x + 6 = 0" ---> True (all quadratics have 2 solutions)


"{-3, -2} is the solution set to (x+3)(x+2) = 0" ---> True ((x+3)(x+2) is equivalent to x^2 + 5x + 6)


"{-3, -2} is the solution set to x+3=0 or x+2 = 0" ---> True (zero product property)


"{-3, -2} is the solution set to x = -3 or x = -2" ---> True (Additive property of equality)


"{-3, -2} is the solution set to -3 = -3 or -2 = -2" ---> True (substution and reflexive property)