Question

Explain the connection between the factors of a polynomial, zeros of a polynomial function, and solutions of a polynomial function.

Answer 1

@jim_thompson5910

Answer 2

well as @satellite73 stated in your last problem, if x-2 is a factor for example, then x = 2 is a root

since

x-2 = 0
x-2+2 = 0+2
x = 2

in general, if x-k is a factor of p(x), then p(k) = 0 basically stating that x = k is a root of p(x)

Answer 3

a "solution" to a polynomial is the same as a root or zero

Answer 4

So the connection would be is that it's always a zero?

Answer 5

yeah if x-3 is a factor, then x = 3 is a zero or root

Answer 6

I don't get it

Answer 7

ok let's say we had `x^2 + 5x + 6 = 0` how would we solve this?

Answer 8

Solve for x

Answer 9

one way is to factor, right?

Answer 10

Yes

Answer 11

what does x^2 + 5x + 6 factor to

Answer 12

-6 and 1?

Answer 13

find two numbers that multiply to 6 and add to 5

Answer 14

3 and 2

Answer 15

-6 times 1 = -6 but we want +6 as a result

Answer 16

3 times 2 = 6
3 plus 2 = 5

works

Answer 17

so x^2 + 5x + 6 factors to (x+3)(x+2)

Answer 18

Oh I get it now

Answer 19

if (x+3)(x+2) = 0 then either x+3 = 0 or x+2 = 0

that leads to x = -3 or x = -2

Answer 20

what @satellite73 was doing in the last problem was thinking backwards from this process to construct a polynomial with the given roots

Answer 21

Oh alright. So the connection between the three things is what?

Answer 22

well in that example, x+3 is a factor so x=-3 is a root

Answer 23

and x = -3 is a solution to the polynomial

Answer 24

But I don't get how that's a connection to factors of polynomials, zeros of polynomials, and solutions of polynomial equation

Answer 25

if you know a zero, then you can find the factor or vice versa

Answer 26

If you answered that question how would you answer it

Answer 27

just like that. That's how I would answer

Answer 28

But I have to explain which sucks haha

Answer 29

So how would you explain it. @jim_thompson5910

Answer 30

I'm basically repeating myself at this point. The explanation is all there. I explained it using an example. You factor x^2 + 5x + 6 to get (x+3)(x+2). Then the factors are used to find the roots using the zero product property

Answer 31

You can take this process in reverse if they said something like "the roots are -3 and -2, find the polynomial"