Question

Decide whether x = −1, x = −3 are zeros of
f(x) = x^4 + 3x^3 + 2x^2 + 6x + 3
1. Only x = −3
2. Both are zeros
3. Only x = −1
4. Neither is a zero

Answer 1

Enjoyed working with you on the previous problem. In this one, why not use synthetic division again to determine whether either -1 or -3 is a root of the given polynomial?

-1 | 1 3 2 6 3
?
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1

Answer 2

how do I determine that?

Answer 3

Have you had any introduction to synthetic division, or any practice with it?

Answer 4

no

Answer 5

Have you had any introduction to long division, or any practice with it?

Answer 6

no

Answer 7

plug in the values x = -1 and x = -3 into f(x) and see if you get 0...

Answer 8

for -1 I don't get 0

Answer 9

then x = -1 is not a zero...what about -3?

Answer 10

i get neither zero?

Answer 11

is that correct?

Answer 12

i agree :)

do you understand *what* a "zero" of a polynomial is?

Answer 13

yes its the intercept right?

Answer 14

x intercept*

Answer 15

sort of...if you're talking about the graph...more specifically, a "zero" of a polynomial is any x-value that makes the polnomial = 0.

they are important because when we want to graph the polynomial, the zeros tell us where it touches (or crosses) the x-axis.

in calculus, they also tell us other things like if the graph is opening up, or down...among other things.

but at the heart of it, if you see the "zero" of a polynomial, really you are just checking (or finding) those values...plugging in is the easiest test method.

Answer 16

mathmale mentioned synthetic division which you will eventually cover...it's useful in finding the zeros because each time you find one, the problem gets easier...