SOLUTION: how would you factor? 2x^4-5x^2-7 please

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Question 1067896: how would you factor?
2x^4-5x^2-7



please

Found 2 solutions by ikleyn, josmiceli:
Answer by ikleyn(52832) About Me  (Show Source):
You can put this solution on YOUR website!
.
how would you factor?
2x^4-5x^2-7

please
~~~~~~~~~~~~~~~~~

Let p(x) = 2x^4 - 5x^2 - 7.

The standard way is THIS:


1.  Introduce new variable u = x^2.   Then your original polynomial takes the form

    q(u) = 2u^2 - 5u - 7.


2.  Find its zeroes. For it, use the quadratic formula

    u%5B1%2C2%5D = %285+%2B-+sqrt%285%5E2+-+4%2A2%2A%28-7%29%29%29%2F%282%2A2%29 = %285+%2B-+sqrt%2825%2B56%29%29%2F4 = %285+%2B-+sqrt%2881%29%29%2F4 = %285+%2B-+9%29%2F4.


    --->  u%5B1%5D = 14%2F4 = 7%2F2,    u%5B2%5D = -1.


3.  So, the decomposition for q(u) is

    q(u) = 2u^2 - 5u - 7 = 2(u-(7/2))*(u-(-1)) = (2u-7)*(u+1).


    Therefore, the decomposition for the original polynomial is

    p(x) = 2x^4 - 5x^2 - 7 = (2x^2-7)*(x^2+1).


    You may factor it further


    p(x) = 2%2A%28x-sqrt%287%2F2%29%29%2A%28x%2Bsqrt%287%2F2%29%29%2A%28x%5E2%2B1%29,   if you want.

Solved.


Answer by josmiceli(19441) About Me  (Show Source):
You can put this solution on YOUR website!
+2x%5E4+-+5x%5E2+-+7+=+0+
make the substitutions:
+z+=+x%5E2+
and
+z%5E2+=+x%5E4+
-------------------
+2z%5E2+-+5z+-+7+=+0+
Use the quadratic formula
+z+=+%28+-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+
+a+=+2+
+b+=+-5+
+c+=+-7+
+z+=+%28+-%28-5%29+%2B-+sqrt%28+%28-5%29%5E2-4%2A2%2A%28-7%29+%29%29%2F%282%2A2%29+
+z+=+%28+5+%2B-+sqrt%28+25+%2B+56+%29%29%2F4+
+z+=+%28+5+%2B-sqrt%28+81+%29%29+%2F+4+
+z+=+%28+5+%2B+9+%29+%2F+4+
+z+=+7%2F2+
and
+z+=+%28+5+-+9+%29%2F4+
+z+=+-1+
-----------------------
+z+=+x%5E2+
+x+=+sqrt%28z%29+
+x+=+sqrt%28+7%2F2+%29+
+x+=+-sqrt%28+7%2F2+%29+
and
+x+=+sqrt%28-1%29+
+x+=+-sqrt%28-1%29+
-----------------------
These are the 4 answers, 2 of which are imaginary
You can check them by plugging into given equation
Here's the plot:
+graph%28+400%2C+400%2C+-4%2C+4%2C+-15%2C+5%2C+2x%5E4+-+5x%5E2+-+7+%29+
Looks close since +sqrt%28+7%2F2+%29+=+1.87+