SOLUTION: list ALL roots (rational, irrational, and/or complex) of the given polynomial equation by using the methods discussed in this lesson. show work 4x^4 - 37x^2 + 9 = 0

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: list ALL roots (rational, irrational, and/or complex) of the given polynomial equation by using the methods discussed in this lesson. show work 4x^4 - 37x^2 + 9 = 0      Log On


   

Question 1206676: list ALL roots (rational, irrational, and/or complex) of the given polynomial equation by using the methods discussed in this lesson. show work
4x^4 - 37x^2 + 9 = 0
Found 2 solutions by MathLover1, greenestamps:
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!
4x%5E4+-+37x%5E2+%2B+9+=+0.... factor
4+x%5E4+%2B+12x%5E3+-+x%5E2+-+3x-12x%5E3+-+36x%5E2+%2B3x+%2B9=0
%284+x%5E4+-12x%5E3%29%2B+%2812x%5E3+-+36x%5E2%29-+%28x%5E2-3x%29-+%283x-9%29=0
4x%5E3%28x+-3%29%2B+12x%5E2%28x+-+3%29-+x%28x-3%29-+3%28x-3%29=0
%28x+-+3%29+%284x%5E3+%2B+12x%5E2+-+x+-+3%29+=+0
%28x+-+3%29+%28%284x%5E3+%2B+12x%5E2%29+-+%28x+%2B3%29%29+=+0
%28x+-+3%29+%284x%5E2%28x+%2B+3%29+-+%28x+%2B3%29%29+=+0
%28x+-+3%29+%28x+%2B+3%29+%284x%5E2-+1%29+=+0
%28x+-+3%29+%28x+%2B+3%29+%28%282x%29%5E2-+1%29+=+0
%28x+-+3%29+%28x+%2B+3%29+%282+x+-+1%29+%282+x+%2B+1%29+=+0

roots:
if %28x+-+3%29++=+0=> x=3
if %28x+%2B+3%29++=+0=> x=-3
if +%282+x+-+1%29++=+0=> x=1%2F2
if +%282+x+%2B+1%29+=+0=> x=-1%2F2

+graph%28+600%2C+600%2C+-10%2C+10%2C+-10%2C+10%2C+4x%5E4+-+37x%5E2+%2B+9+%29+


Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


4x%5E4-37x%5E2%2B9

The polynomial will factor in the form

%28ax%5E2-b%29%28cx%5E2-d%29

where ac=4 and bd=9.

Trying different combinations, we easily find

4x%5E4-37x%5E2%2B9=%284x%5E2-1%29%28x%5E2-9%29=%282x%2B1%29%282x-1%29%28x%2B3%29%28x-3%29

The roots are

ANSWER: -1/2, 1/2, -3, 3