SOLUTION: Check whether x+√3 is a factor of x^4+x^2-12

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Question 1037258: Check whether x+√3 is a factor of x^4+x^2-12
Answer by Theo(13342) About Me  (Show Source):
You can put this solution on YOUR website!
you can use synthetic division to prove that it is a factor.

take x^4 + x^2 - 12 and fill in the missing terms in descending order of degree to get:

x^4 + 0x^3 + x^2 + 0x - 12.

take the coefficients of each of these terms to get:

1 + 0 + 1 + 0 - 12

take the factor of x + sqrt(3) and set it equal to 0 and solve for x to get:

x = -sqrt(3).

your synthetic division will be -sqrt(3) being synthetically divided into 1 + 0 + 1 + 0 - 12.

see the following worksheet for the details of the calculations.

$$$

if you don't know how to do synthetic division, then check out the following tutorial.

http://www.purplemath.com/modules/synthdiv.htm

you could also have done the following:

solve for x + sqrt(3) = 0 to get x = -sqrt(3).

since f(x) = x^4+x^2-12, then replace x with -sqrt(3) to get:

f(-sqrt(3)) = (-sqrt(3))^4 + (-sqrt(3))^2 - 12.

evaluate this equation to get f(-sqrt(3)) = 0.

if f(a) is equal to 0, then a is a root and (x-a) is a factor.

you can also solve this graphically by graphing the equation of y = x^4 + x^2 - 12.

your graph will look like this:

$$$

the graph shows that the roots of the equation are plus and minus 1.732.

sqrt(3) is equal to 1.732050808.

plus and minus 1.732 are just rounded versions of plus and minus sqrt(3).

the factors of x^4 + x^2 - 12 are (x^2 + 4) * (x^2 - 3).

set these factors to 0 and solve for x to get:

x^2 + 4 = 0
subtract 4 from each side of the equation to get x^2 = -4
take the square root of both sides of this equation to get x = plus or minus sqrt(-4).
those are imaginary roots (also called complex), and therefore don't show up on the graph.

x^2 - 3 = 0
add 3 to both sides of the equation to get x^2 = 3
take the square root of both sides of this equation to get x = plus or minus sqrt(3).
those are real roots, and therefore do show up on the graph.

if you look at the factor of (x^2 - 3), this can be simplified further into (x + sqrt(3)) * (x - sqrt(3)).

that's where the factor of (x + sqrt(3)) came from.