Are there any methods regarding factoring higher order polynomials with prime factors but not real solutions (as this makes using the Polynomial Remainder Theorem useless)?
For example:
P(x) = x⁴+64
Can be factored as
P(x) = (x²-4x+8)(x²+4x+8)
Or
P(x) = 2x^5 + 2x⁴ - x³ + 2x² - 1
Which can be factored as
P(x) = (2x³+x+1)(x²+x-1)
I guess you could try factoring into two complete polynomials with unknown coefficients and solve the equations to find the values but that seems too tedious.
The only other way I came up with was plugging in values for x until I found one that could be expressed as the multiplication of two primes, hoping ir would give me the correct answer.
For example, with
P(x) = 2x^5 + 2x⁴ - x³ + 2x² - 1
I did
P(0) = -1
P(1) = 4
P(2) = 95
Since 95 = 19*5, I just kind of forced x to fit that pattern
So P(2) = (2² + 2 - 1)(2*2³ + 2 + 1)
And P(x) = (x²+x-1)(2x³+x-1)
Now, I was lucky that this gave me the correct answer (I checked it by multiplying the two prime factors) because I was basicallly just looking for a pattern in the dark.
For example:
P(x) = x⁴+64
Can be factored as
P(x) = (x²-4x+8)(x²+4x+8)
Or
P(x) = 2x^5 + 2x⁴ - x³ + 2x² - 1
Which can be factored as
P(x) = (2x³+x+1)(x²+x-1)
I guess you could try factoring into two complete polynomials with unknown coefficients and solve the equations to find the values but that seems too tedious.
The only other way I came up with was plugging in values for x until I found one that could be expressed as the multiplication of two primes, hoping ir would give me the correct answer.
For example, with
P(x) = 2x^5 + 2x⁴ - x³ + 2x² - 1
I did
P(0) = -1
P(1) = 4
P(2) = 95
Since 95 = 19*5, I just kind of forced x to fit that pattern
So P(2) = (2² + 2 - 1)(2*2³ + 2 + 1)
And P(x) = (x²+x-1)(2x³+x-1)
Now, I was lucky that this gave me the correct answer (I checked it by multiplying the two prime factors) because I was basicallly just looking for a pattern in the dark.
