Really Important Factoring Question?
The expression 3x^2 - 11x + k can be factored into two linear polynomials with integer coefficients. Determine the possible values of k.
Can someone do this step by step please.
- 1 decade agoFavorite Answer
Hi, first of all notice that the given polynomial can be factored as a product of linear factors with integer coefficients if and only if it has rational roots a/b and c/d such that a/b + c/d = 11/3 and (a/b)(c/d) = k/3 where either a = b = 1 or one of them is 3 and the other is 1 (this comes from the fact that the polynomial factors in 3(x - a/b)(x - c/d) and 3 is prime). Assuming a = b = 1 you get the system
a + c = 11/3
ac = k/3
which has clearly no integer solutions. Then you must have for example b = 3 and d = 1 (the choice will not affect the result) and in this case the system becomes
a/3 + c = 11/3
ac = k,
a + 3c = 11
ac = k
with a and c integers. From the first one you get a = 11 - 3c, which substituted in the second one gives you
k = c(11 - 3c),
which as c varies in Z should give you all the possible values of k for which 3x^2 - 11x + k can be factored into linear polynomials with integer coefficients.
I hope this helped you. Bye!
@Roger The Mole: for any integer c if k = c(11 - 3c) the discriminant is
36c^2 - 132c + 121 = (6c - 11)^2,
that is a perfect square. So for example k = -70 (you get it or c = 10) is a good value, which leads to the decomposition
(3x + 10)(x - 7).
Moreover, the polynomial p(x) = x is linear, it's not suspicious, it suits the definition.
- Roger the MoleLv 71 decade ago
For the expression to be factorable in integers, there must be integer solutions for
3x^2 - 11x + k = 0.
Since this is a quadratic equation, that means that the discriminant must be a perfect square. That is, (121 - 12k ) must be a perfect square.
That is true for k = -4, 0 , 6, 8, and 10.
The solution k = 0 is suspicious, because x = 0 might not be considered one of "two linear polynomials".
The corresponding factorizations, including the case k = 0, are:
(x - 4) (3x + 1)
x (3x - 11)
(x - 3) (3x - 2)
(x - 1) (3x - 8)
(x - 2) (3x - 5)