How do I factor this equation?

Question

x⁴ +6x² - 32x + 25

Answer 1

x⁴ +6x² - 32x + 25

 

You have ax⁴ + bx³ + cx² + dx + e

a = 1, b=0, c = 6, d = -32, e = 25

 

The roots of the polynomial will be found by

the factors of the constant, 25, divided by the

factors of the coefficient of the highest term

which is 1 from the a=1 on x⁴

 

Your initial choices for roots are the roots of 25/1

25/1 = 25

The possible roots to try are 1,-1, 5, -5, 25, -25

 

If 1 is a root then dividing by (x-1) will give no remainder.

 

Use synthetic division with the coefficients

1, 0, 6, -32, 25

 

1  |  1   0    6    -32    25

    |      1     1       7  -25

    ---------------------------

       1   1    7     -25    0

 

The remainder is 0 so 1 is a root and the polynomial is

divisible by (x-1) leaving  1x³ + 1x² + 7x -25

 

The factor is:

 

(x-1)(x³ + X² + 7X - 25)

 

 Given that the polynomial is 4th order, we

expect to find 4 roots.  However, there is

no guarantee that the roots are rational or real.

 

Looking at (x³+x²+7x-25) and following the

same usage of the rational roots theorem and

synthetic substitution we need to check roots

for the values of -25/1

Thus again we have possible roots of 1, -1, 5, -5, 25, -25

 

However... none of those work to a remainder of zero

so we assume the other 3 roots are imaginary or

irrational and we are done at the point:

(x-1)(x³ + X² + 7X - 25) 

 

 

Answer 2

Using the Rational Root Theorem  and Synthetic substitution:

 

(x - 1)(x³ + x² + 7x - 25)