Question

f(x)=(x-1)^3(x+3)(x-4)^2. Determine how to find the x and y intercepts and the multiplicity of zeros.

Answer 1

well, when x=0, what value do we obtain for y?

and the zeros refer to the values of x that make any factor equal out to zero ...

Answer 2

the exponents are shorthand for how many times a factor appears, so by default they also define the number of times (multiplicity) a value of x produces a zero

Answer 3

I got -48 for y

Answer 4

(-1)^3 (+3) (-4)^2
-1 3 16 ; -48 for y intercept is good

Answer 5

now we should know by now that:

x - r = 0, when x=r so what would you say are the zeroes?

Answer 6

Im sorry where are you getting r?

Answer 7

r is some random general variable expressing any value whatsoever that you can think of as opposed to just the specific 3 values presented in the problem.

x - r = 0 , anytime x = r, therefore; x-3 = 0 when x=3



i good trick is simply to swap signs on that y intercept parts

from: (-1)^3 (+3) (-4)^2
to: (+1)^3 (-3) (+4)^2

this defines our zeros AND the exponents define the multipicities

Answer 8

so the zeros would be 1, -3 and 4?

Answer 9

yes :) and the exponents attached to them are their multiplicities

Answer 10

1^3 = 1*1*1 , so a multiplicity of 3

-3^1 = -3 , so a multiplicity of 1

4^2 = 4*4 , so a multiplicity of 2

Answer 11

so I have the y intercept as -48 but the x intercept I would set the equation equal to zero right?

Answer 12

yes, for x intercepts you would set the equation to zero.

and since it is already in a factored form, and you are just mulitplying things together, we should know by now that anything times 0 is equal to 0.

which is why its simplicity itself to say that the roots, or zeros, are such that "x = r"

Answer 13

ok

Answer 14

so it would be 0=(x-1)^3(x+3)(x-4)^2

Answer 15

correct

Answer 16

can you help me with the first step in solving that?

Answer 17

i already have.

spose we rewrite it as:

0 = a*b*c ; what would you say would fit this equation?

Answer 18

0

Answer 19

if a = 0, it is true since 0 = 0*b*c
if b = 0, it is true since 0 = a*0*c
if c = 0, it is true since 0 = a*b*0

therefore, we can assess each factor and equate them to zero.

Answer 20

0 = (x-1)^3 (x+3) (x-4)^2

when x-1 = 0, solve for x
when x+3 = 0, solve for x
when x-4 = 0, solve for x

Answer 21

oh so the zeros are the x intercepts?