Factor this polynomial:
F(x)=x^3-x^2-4x+4
Try to find the rational roots. If p/q is a root (p and q having no factors in common), then p must divide 4 and q must divide 1 (the coefficient of x^3).
The rational roots can thuis be +/1, +/2 and +/4. If you insert these values you find that the roots are at
x = 1, x = 2 and x = -2. This means that
x^3-x^2-4x+4 = A(x - 1)(x - 2)(x + 2)
A = 1, as you can see from equation the coefficient of x^3 on both sides.
Typo:
The rational roots can be
+/-1, +/-2 and +/-4
To factor the polynomial F(x) = x^3 - x^2 - 4x + 4, we can start by trying to find the rational roots using the rational root theorem.
The rational root theorem states that if a polynomial has a rational root p/q, where p is a factor of the constant term and q is a factor of the leading coefficient, then p/q is a possible root of the polynomial.
In this case, the constant term is 4 and the leading coefficient is 1. So the possible rational roots can be obtained by dividing factors of 4 by factors of 1, which gives us: ±1, ±2, ±4.
By trying these possible roots, we can determine the actual roots of the polynomial.
If we substitute x = 1 into F(x), we get F(1) = 1^3 - 1^2 - 4(1) + 4 = 0. Therefore, x = 1 is a root.
If we substitute x = 2 into F(x), we get F(2) = 2^3 - 2^2 - 4(2) + 4 = 0. Therefore, x = 2 is a root.
If we substitute x = -2 into F(x), we get F(-2) = (-2)^3 - (-2)^2 - 4(-2) + 4 = 0. Therefore, x = -2 is a root.
Now that we have determined the roots of the polynomial, we can write it in factored form. Since the leading coefficient is 1, we can write the equation as:
F(x) = 1(x - 1)(x - 2)(x + 2)
This is the factored form of the polynomial F(x).