factor
## Zero Product Property

in this lesson we're going to focus on

the zero product property

and the basic idea behind the zero

product property

is

if you're multiplying two numbers a and

b

and if a and b

equals zero

then either a or b must be zero

because zero times anything

is zero

now let's think about this let's say we

have two numbers that multiply

to zero

the only way this could happen is if one

of those numbers is zero

for instance if we have eight

if the first number is eight the second

number has to be zero

it really doesn't matter what the first

number is if it's 6 12 15 as long as the

second number is zero the whole thing is

going to be zero

if the first number is zero it doesn't

matter what the second number is it

could be negative four seven

twelve

0 times anything is zero

so whenever you have two things that

multiply to zero

one of those things

has to be zero

now this property is very useful

when solving quadratic equations

typically

when you factor a trinomial

you might get something that looks like

this

and you need to solve for x

using a zero product property we can do

that

if either x minus three or x plus two is

equal to zero

this entire expression will be equal to

zero

so using the zero product property

we can break this single equation into

two parts

we can set x minus three equal to zero

and we can set x plus two equal to zero

because if just one of these

two factors equals zero the whole thing

is going to be zero

and so that's why we can do this it's

because of the zero product property

now once we have these two equations we

can go ahead and solve it

to get the answer for the first one we

just got to add 3 to both sides

and we'll get x is equal to positive 3.

for the second one

we need to subtract both sides by 2

and we'll get x is equal to negative 2.

so that's one of the applications

of using

the zero product property

it's very useful when solving quadratic

equations especially when you're

factoring

for the sake of practice

go ahead and calculate the value of x

for these two equations

the first one is going to be 3x

times x minus seven is equal to zero

and for the second one

it's going to be two x minus three

times three x minus five equal to zero

so use the zero product property

to calculate the value of x for each of

these two equations

now for the first one

what we can do is we can set each factor

equal to zero

so we can set 3x equal to zero

and x minus seven equal to zero

for the first one we need to divide both

sides by three

three x divided by three is simply x

zero divided by three is zero

so the first answer is just x is equal

to zero

for the second equation

we simply need to add seven

to both sides

and we get the answer x is equal to

seven

so if we were to plug in 0

or 7

into the original equation

it's going to work

for instance if we plug in 0

into each x value we'll have three times

zero

times zero minus seven

three times zero is zero

zero minus seven is negative seven

zero times negative seven is zero

so this works

now if we plug in seven

it will work as well

three times seven

and then seven minus seven that's gonna

be zero three times seven is twenty one

seven minus 7 is 0

21 times 0 is 0.

so using the zero product property

we can solve for x

whenever it's in factored form

now let's try the other example

now we're going to follow the same

process

we're going to set each factor equal to

zero

so we'll break it up into two equations

two x minus three is equal to zero

and three x minus five is equal to zero

so let's begin by adding three

to both sides

so we'll get 2x is equal to 3

and then we'll divide both sides by 2.

so the first answer that we get

is x is equal to three over two

if we were to plug this in to the

original equation

and this part will equal zero which

means the whole equation will equal zero

now for the second one we're going to

add five

to both sides

so we'll get 3x is equal to 5

and then we'll divide both sides by

3.

so we'll get x is equal to 5 over 3.

so if you were to plug in any one of

these two x values into the original

equation

the whole thing is going to equal 0.

so that's how you can solve equations

using the zero product property

first it needs to be in factored form

like this

and then you could set each factor equal

to zero

and then solve for x

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