can someone please explain rational root theorem
In: Mathematics
If you have a polynomial with integer coefficients and a nonzero constant, and that polynomial has a root that is rational, then the constant term will be divisible by the root’s numerator and the leading coefficient will be divisible by the root’s denominator (assuming the root is in simplest terms). For example,
3x^2 + 2x – 1
This is a polynomial with integer coefficients and a nonzero constant. By the theorem, the rational roots of this polynomial (if it has any) must have a numerator of 1 or -1 (since those are the only divisors of the constant term) and a denominator of 3, -3, 1, or -1 (since those are the only divisors of the leading coefficient). Thus, the only possible rational roots this polynomial could have are 1/3, -1/3, 1, and -1.
Plugging in 1/3 and -1, you’ll find that these are the roots of the polynomial (it only has two since the highest power is 2), while plugging in -1/3 and 1 will show that those possibilities are not roots.
From this, you can factor the polynomial as
3x^2 + 2x – 1 = 3(x-1/3)(x+1).
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