Question

If t is an unknown constant which binomial must be a factor of 7m(squared) + 14m - 2t

Answer 1

I'd start by applying the quadratic formula. Here, I'd identify each of the coefficients a, b and c from the given expression: a=7, b=14, and c=-2t.

Substitute these into the quadratic formula, solving for m:\[m=\frac{ -14.plus.or.minus \sqrt{14^{2}-4(7)(-2t)} }{ 2(7) }.\]
Now if we were to let the "discriminant," 14^2-4(7)(-2t) = 0 and solve for t, we'd get 3.5 for t. Then, with t=3.5, the roots of the quadratic would be \[m=\frac{ -14.plus.or.minus.zero }{ (2)(14)}.\]

Answer 2

We'd then have two equal, real roots, both equal to -14/[(2)(14)], or -1/2.

Thus, factors of the quadratic would be (m+(1/2)), at least for t=3.5.

What if the discriminant were not zero, but also not negative? Then the roots m would be both real and different. Could we identify a more general binomial factor of the given expression that includes the variable t and thus predicts the binomial factor in terms of the value of t?