Question

What polynomial has roots of −6, −4, and 1?

Answer 1

You can build a polynomial with factors. For instance, a polynomial with roots 2 and 3:

x = 2, 3
(x - 2)(x-3) = 0
x^2-5x+6 = 0

Answer 2

oh okay

Answer 3

how does that work? like, the roots and stuff?

Answer 4

a factor when root is 3 will be (x - 3) Because when x - 3 = 0 x = 3

if root is -2 the factor is ( x + 2) Because when x + 2 = 0 x = -2

Answer 5

so a polynomial with roots -2 and 3 is (x + 2)(x - 2)

Answer 6

So do same process with -6,-4 and 1. You will have 3 factors

Answer 7

oh okay, thank you!

Answer 8

it makes so much sense now XD

Answer 9

yw

Answer 10

I have (x+6) (x+4) (x-1)

Answer 11

that's right, you have to expand it for it to be a polynomial, though.

Answer 12

okay

Answer 13

I foil?

Answer 14

@Lord_Box

Answer 15

Yes

Answer 16

First do
(x+6) (x+4)
x^2 + 4x+6x+24
or
x^2 +10x+24
now multiply that by (x-1)
one way is to distribute (x-1)

for example
(x-1)(x^2 +10x+24)
means multiply *each* term inside the parens by (x-1)
(x-1)x^2 + (x-1)*10x + (x-1)*24
now simplify each term (distribute again!) and collect terms
can you do that ?

Answer 17

ok

Answer 18

(x^3-1x^2) + (10x^2-10x) + (24x-24) ?

Answer 19

That's right, and combine your like terms.

Answer 20

I got x^3+9x^2+14x-24

Answer 21

looks good

Answer 22

now you can try to "go backwards" and try to figure out the roots of that equation
i.e. what x values make it zero. That is a much harder problem (but one you will often do for quadratic i.e. power of 2 (not 3 like here))

Answer 23

okay

Answer 24

so i start like this? x^3+9x^2+14x-24= 0

Answer 25

yes, but it's a hard problem (I don't think people even teach how to do it by hand)
Don't bother unless you are a math nerd.

Answer 26

What should i do now then? I still have to figure out the answer

Answer 27

the question is
**What polynomial has roots of −6, −4, and 1?**
the answer is what you found
x^3+9x^2+14x-24

Answer 28

Terminology wise:
x^2 is a monomial
x^2 + x is a binomial and a polynomial
x^2 + x + 1 is a trinomial and a polynomial.

polynomials have multiple terms

Answer 29

oh, hold up

Answer 30

the answer choices are
3x2 + 16x + 42
3x2 + 4x + 2
3x2 − 16x + 42
3x2 − 4x + 2

Answer 31

those are the choices to what question ?

Answer 32

The original one *What polynomial has roots of −6, −4, and 1?*

Answer 33

A quadratic equation can have at most 2 roots

Answer 34

First, if a polynomial has 3 roots, it has "order 3" i.e. the highest exponent must be 3
(as in we *need* an \( x^3\) term
Second, if we sub in x=1 into all 4 choices, none give 0

So we have a problem here. Mention it to your teacher.

Answer 35

okay, this is pretty confusing. I'll guess on this one for now, thanks for the help!

Answer 36

You probably have a typo
or possibly you are misreading the question.
Can you post a screen shot ?

Answer 37

sure, hold up

Answer 38

ok
the question is
**What polynomial has roots of −6, −4, and 1?**
the answer is
x^3+9x^2+14x-24

Answer 39

one of the 4 choices is the answer.

Answer 40

oh wait...

Answer 41

i must be really tired XD

Answer 42

Thank you!