Question

Please help!! Which of the following is a polynomial with roots 4, −5, and 7?

f(x) = x3 − 6x2 − 27x + 140

f(x) = x3 − 6x2 − 20x + 27

f(x) = x3 − 20x2 − 27x + 35

f(x) = x3 − 20x2 − 35x + 140

Answer 1

\(\large\color{black}{ \displaystyle {\rm~A~polynomial~with~root~}a~{\rm is}:}\)
\(\large\color{black}{ \displaystyle f(x)=x-a}\) (A line)


Same way,

\(\large\color{black}{ \displaystyle {\rm~A~polynomial~with~roots~}a,~{\rm and}~b~{\rm is}:}\)
\(\large\color{black}{ \displaystyle f(x)=(x-a)(x-b)}\)


AND, same way

\(\large\color{black}{ \displaystyle {\rm~A~polynomial~with~roots~}a,~b,~{\rm and}~c~{\rm is}:}\)
\(\large\color{black}{ \displaystyle f(x)=(x-a)(x-b)(x-c) }\)

And so on

Answer 2

But, these are polynomials "of the least degree" and with "scale factor" of 1.

Answer 3

It seems like your polynomials are indeed least degree polynomial.
So just you my first post to find your polynomial.

Answer 4

So how would I get x. Sorry I'm really confused:(

Answer 5

??

Answer 6

How would I start the problem? I read the lesson, but I don't get it

Answer 7

See my first post?
Look at the last polynomial with roots a, b, and c.

The reason that this works, because my f(x)=0,
when evaluated at x=a, b or c.
((Because 0 times anything is zero))

So you just model the same thing,
but your a, b, and c are:
4, -5, and 7

(The order doesn't matter)

Answer 8

`STEP 1:`
Model the f(x)

((just as I showed, WITH a, b, and c
BUT, YOU USE YOUR NUMBERS))

`Step 2`
Then, EXPAND your polynomial.

Answer 9

(x-4)(x-4)(x-7)? So then I would expand?

Answer 10

check parenthesis #2, please.

Answer 11

whoops. (x-4)(x-5)(x-7)?

Answer 12

your second zero is NEGATIVE 5.

Answer 13

(\(x-\color{red}{-5}\))=(x+5)

Answer 14

(x-4)(x+5)(x-7)

Answer 15

Yes, then expand :)

Answer 16

So then I'd multiply?

Answer 17

Yes, expand it...

Answer 18

Would it be (x) = x^3 − 20x^2 − 35x + 140

Answer 19

only the +140 and x^3 are correct...

NO, it would not.

Answer 20

next time, please show your work.

Answer 21

Ok, what is
(x-4)(x+5) ?

Answer 22

(without showing work, I can't tell you where you made the error in expanding)

Answer 23

x^3 − 6x^2 − 27x + 140

Answer 24

Yes, perfect!

Answer 25

I understand now:) Thanks!

Answer 26

Alrighty ... yw