list all the possible integral zeros:
P(x)= x^3-2x^2-11x+12
The factor theorem helps us to say that the possible integral roots are in the form of (x±k), where k is a possible factor of the constant term (12) divided by a possible factor of the leading coefficient (1).
Thus for the given problem, the possible integral zeroes are
±1,±2,±3,±4,±6,±12.
Also, the sign of the coefficients changed twice, so by Descartes rule of signs, there are zero or two positive roots, and one negative root.
To find the possible integral zeros of a polynomial, we can use the Rational Root Theorem. According to the theorem, the possible integral zeros of a polynomial with integer coefficients can be determined by using the factors of the constant term (in this case, 12) divided by the factors of the leading coefficient (in this case, 1).
The factors of the constant term 12 are ±1, ±2, ±3, ±4, ±6, and ±12.
The factors of the leading coefficient 1 are ±1.
So, the possible integral zeros are the potential combinations of these factors:
±1, ±2, ±3, ±4, ±6, ±12
To determine which of these are actual zeros, we can use polynomial long division or synthetic division.
To find the possible integral zeros of a polynomial, we can apply the Rational Root Theorem. According to the theorem, if a polynomial has any rational zeros, they must be in the form of p/q, where p is a factor of the constant term and q is a factor of the leading coefficient.
Let's find the constant term and the leading coefficient of the polynomial P(x) = x^3 - 2x^2 - 11x + 12. The constant term is 12, and the leading coefficient is 1.
Now, let's find the factors of the constant term (12): ±1, ±2, ±3, ±4, ±6, ±12.
And the factors of the leading coefficient (1): ±1.
To determine the possible rational zeros, we can form all possible combinations of these factors. This means we need to divide the factors of the constant term by the factors of the leading coefficient, using both positive and negative forms.
The possible rational zeros are:
±1/1, ±2/1, ±3/1, ±4/1, ±6/1, ±12/1.
Simplifying these fractions, we get:
±1, ±2, ±3, ±4, ±6, ±12.
Therefore, the possible integral zeros of the polynomial P(x) = x^3 - 2x^2 - 11x + 12 are:
-12, -6, -4, -3, -2, -1, 1, 2, 3, 4, 6, 12.