Question

Determine the x intercepts, zeros, and factors for the polynomial f(x)= 3x^4+3x^3-18x^2

Answer 1

each term has a common factor of \(3x^2\) so the first step is to factor it out and write
\[f(x)=3x^2(x^2+x-6)\]

Answer 2

next step is to factor \(x^2+x-6\)
do you know how to do that?

Answer 3

yes?

Answer 4

ok, what do you get?

Answer 5

x=2?

Answer 6

wait no 3

Answer 7

i mean "what to you get when you factor?"

Answer 8

more precisely, what do you get when you factor \(x^2+x-6\) ?

Answer 9

I get the x intercepts

Answer 10

right?

Answer 11

or the zero's

Answer 12

you get the factors that you set equal to zero to find the intercepts, yes

Answer 13

but my question is "how did you factor \(x^2+x-6\) ? " what are the factors?

Answer 14

positive 6 ? wait okay, I add 6 to 0 and then what do I do with the x^2

Answer 15

ok lets go slow

Answer 16

you want to factor \(x^2+x-6\) so you have to think of two numbers whose product is \(-6\) and whose sum is 1
i.e. to numbers that, when you multiply them together you get \(-6\) and when you add them you get \(1)
since -6 is negative, one of them will be positive and the other will be negative

Answer 17

-3 2

Answer 18

those are the zeros, yes, but actually it factors as
\[x^2+x-6=(x+3)(x-2)\]

Answer 19

because you switch em!

Answer 20

the plus and minus

Answer 21

so if \(x+3=0\) then \(x=-3\) and if \(x-2=0\) then \(x=2\)

Answer 22

okay! and that's the zero"s?

Answer 23

so you have three zeros

Answer 24

they are \(x=2, x=-3\) and also don't forget \(x=0\)

Answer 25

x=-3
x=2 and x=0

Answer 26

yes

Answer 27

okay what about the x intercepts

Answer 28

those are the x intercepts

Answer 29

x intercept is a synonym for "zeros"

Answer 30

they are the same thing??

Answer 31

yes

Answer 32

are the factors of the polynomial the same thing as well?

Answer 33

if \(f(2)=0\) that means \((0,2)\) is on the graph, i.e. that is where the graph crosses the \(x\) axis

Answer 34

not exactly

Answer 35

\[f(x)= 3x^4+3x^3-18x^2=3x^2(x+3)(x-2)\] the "factors" are
\[3x^2, x+3, x-2\]

Answer 36

the "zeros" are \(\{0,-3,2\}\)

Answer 37

the "x intercepts" are \((0,0), (-3,0), (2,0)\)

Answer 38

oh okay, I was thinking the x intercepts where something different but I understand now

Answer 39

but clearly they are related

Answer 40

thanks so much

Answer 41

yw