Question

Which of the following is a polynomial with roots negative square root of 3, square root of 3, and −2?

x3 − 2x2 − 3x + 6
x3 + 2x2 − 3x − 6
x3 − 3x2 − 5x + 15
x3 + 3x2 − 5x − 15

Answer 1

You are given the three roots
\[x=\sqrt{3}, x=-\sqrt{3}, x=-2\]

So you can just substitute these into the given polynomials, if the answer is 0 then it is a root of that polynomial

for example

\[x^3 - 2x^2 -3x +6\]
let \[x=\sqrt{3}\]

then

\[\sqrt{3}^{3} -2(\sqrt{3})^2 -3\sqrt{3} + 6 = 3\sqrt{3} - 6 - 3\sqrt{3} + 6 = 0\]

then repeat by letting x equal the other two roots, and continue to substitute them into all of the given polynomials, until a polynomial gives 0 as the answer for all three roots

Answer 2

I dont quite understand..

Answer 3

@amorfide

Answer 4

I believe y our best bet is to form the three factors of this polynomial and to multiply them together. That will most positively help you choose the correct answer.

If x=-2 is a root, then (x+2) is a factor. See that sign change?

If \[x=\sqrt{3}\]

Answer 5

is a root, then \[(x-\sqrt{3}) \]

Answer 6

I am so lost.

Answer 7

is a factor. If negative sqrt 3 is a root, \[x=-\sqrt{3}\]

Answer 8

is a factor. If neg sqrt (3) is a root, what is the 3rd factor?

Answer 9

I'm sorry you find this to be difficult to grasp.
Could you possibly ask questions?

Let me work out one sample problem for you.

Given that x=2 and x=3 are roots of a quadratic equation, write the actual quadratic.

Answer 10

Could you work this one out lol, theres another just like it that i have to do using the same numbers

Answer 11

Take that x=2 and write the factor (x-2); take that x=3 and write the factor (x-3). You really do have to know and undrstand this fact.

Answer 12

What is the polynomial? Use the FOIL method to multiply: (x-2)(x-3).

Can you do that?

Answer 13

First terms: x times x = x^2 ("x squared")

Answer 14

Outer terms: (x)(3) = 3x

Answer 15

Inner terms: (-2x)(-3x) = ?

Answer 16

x^2-3x-2x+6

Answer 17

Yes, and then combine the 2 middle terms. What do you get?

Answer 18

x2-5x+6

Answer 19

Perfect, except that you should write the first term as x^2, or as \[x^2.\]

Answer 20

correct

Answer 21

What is your final answer? In other words, where is the 3rd order polynomial you've created?

Answer 22

Looks like I may have misled you.

If you have roots POS Sqrt(3), NEG Sqrt(3), and -2, write the appropriate factors:\[(x+\sqrt{2})(x-\sqrt{2})(x+2)\]

Answer 23

Sorry for the extra work. Pls multiply out these 3 factors and simplify.\[(x-\sqrt{3})(x+\sqrt{3})(x+2).\]

Answer 24

Important hint:

Answer 25

(a-b)(a+b)=a^2-b^2