This is a good question and does not have a general answer!
Let's call p(x) a polynomial in x of degree n>=0 an expression of the form.
(If n=0 the polynomial is called constant)
p(x) = Sum(a_i * x^i, i=0..n),
this is a sum of terms a_i * x^i where i ranges from 0 to n.
A factor of a polynomial p(x) is another polynomial q(x) that divides it. (Since we can define division among polynomials, it turns into a ring. But that's beyond this answer). For instance if there is a polynomial r(x) such that p(x) = q(x)*r(x), then p(x)/q(x) = r(x). This is pretty straightforward. Note that divisors of a polynomial can have any degree, hence do not have to have degree 1.
Mathematicians have proved that polynomials of degree 1, 2, 3, 4, 6, 8 can be factored with reasonable methods, but if the degree is 5, 7 or higher than 8 there is no general method. I'll give another positive result later on.
The reason for this is the following.
Factoring polynomials has a relation with finding its roots.
A root (or zero) of a polynomial p(x) is a value of x such that p(x) = 0.
Now suppose that p(x)=0 and p(x) can be factored as p(x) = q(x)*r(x), then it follows that q(x) = 0 or r(x) = 0 and we found a zero of one of its divisors. We can turn this around too: if p(a) = 0 for some a, then there are polynomials q(x) and r(x) such that p(x)/q(x) = r(x). In particular q(x) = x-a can be taken. (I won't prove it here.)
This gives you one method to find a factor of a polynomial: find a zero of the polynomial.
The main theorem of algebra is this:
Every polynomial has a complex zero.
A complex number is of the form x + i.y, where
i = (-1)^(1/2) is the root of X^2 + 1.
A complex number is always the root of a second degree polynomial with real coefficients a_i (see definition of polynomial). (I won't prove that here).
This gives you another method to find a factor: find a complex root of p(x) and find the second degree polynomial that it is a root of. This will be a factor of p(x).
Example: Factor X^4 + 1.
Answer: [X^2 - X*{2^(1/2)}+ 1]*[X^2 + X*{2^(1/2)}+ 1]
Note that only real numbers appear in this expression!
To check the answer, just multiply the factors to arrive at the original polynomial.
I like to give you the derivation of this, since it will show you how the complex roots work together to find the factors of X^4 + 1.
Note that the factors of this polynomial lie on a circle of radius one, and form a square with points {(1/2)*2^(1/2)*(1+i), (1/2)*2^(1/2)*(1-i), (1/2)*2^(1/2)*(-1+i); (1/2)*2^(1/2)*(-1-i)}.
For better understanding, copy this and change the expression (1/2)*2^(1/2) as half of the square root of 2.
This actually means that
X^4 + 1 =
[X - (1/2)*2^(1/2)*(1+i)]*
[X - (1/2)*2^(1/2)*(1-i)]*
[X - (1/2)*2^(1/2)*(-1+i)]*
[X - (1/2)*2^(1/2)*(-1-i)]
Now multiply the first two factors and the last two factors.
First note that (1+i)*(1-i) = 1 - i^2 = 1 - (-1) = 2 to find
X^4 + 1 =
[X^2 - X*{(1/2)*2^(1/2)*(1-i)+(1/2)*2^(1/2)*(1+i)}+ 2*{(1/2)*2^(1/2)}^2]
[X^2 - X*{(1/2)*2^(1/2)*(-1+i)+(1/2)*2^(1/2)*(-1-i)}+ 2*{(1/2)*2^(1/2)}^2]
=
[X^2 - X*{2^(1/2)}+ 1]
[X^2 + X*{2^(1/2)}+ 1]
The GCF. Or the greatest common factor.
Too bad that's not a^2 - ab - 42b^2 That factors to (a + 6b)(a - 7b)
In algebra, the factor theorem is a theorem linking factors and zeros of a polynomial. It is a special case of the polynomial remainder theorem.The factor theorem states that a polynomial has a factor if and only if
Factor it once, and then factor the factors.
True
This polynomial doesn't factor. The only thing you can do is take out parts of some terms, e.g. 2(2x3 + 10x2 + x) - 3.
It is an algebraic factor.
(5a2 - 3)(5a2 + 3)
You can factor a polynomial using one of these steps: 1. Factor out the greatest common monomial factor. 2. Look for a difference of two squares or a perfect square trinomial. 3. Factor polynomials in the form ax^2+bx+c into a product of binomials. 4. Factor a polynomial with 4 terms by grouping.
(x + 5) (x - 4)
If there is no common factor then the polynomial cannot be factorised. If there is no common factor then the polynomial cannot be factorised. If there is no common factor then the polynomial cannot be factorised. If there is no common factor then the polynomial cannot be factorised.
Factor the polynomial x2 - 10x + 25. Enter each factor as a polynomial in descending order.
(y-3 z) (9 y+2 z)
-2(2x^4 - 13x^3 + 15)
Too bad that's not a^2 - ab - 42b^2 That factors to (a + 6b)(a - 7b)
In algebra, the factor theorem is a theorem linking factors and zeros of a polynomial. It is a special case of the polynomial remainder theorem.The factor theorem states that a polynomial has a factor if and only if
n^2(n + 6)(n^2 - 6n + 36)
In algebra, the factor theorem is a theorem linking factors and zeros of a polynomial. It is a special case of the polynomial remainder theorem.The factor theorem states that a polynomial has a factor if and only if