If you ever have to solve a quadratic-like trinomial, you're in luck; this type of equation is a perfect candidate for factoring and then for the application of the multiplication property of zero.
A quadratic-like trinomial is a trinomial of the form ax2n + bxn + c = 0. The power on one variable term is twice that of the other variable term, and a constant term completes the picture.
The multiplication property of zero states that if the product of
then at least one of the factors has to represent the number 0.
Now, try an example: Solve the trinomial z6 – 26z3 – 27 = 0.
You can think of this equation as being like the quadratic
x2 – 26x – 27,
which factors into
(x – 27)(x + 1).
If you replace the x's in the factorization with z3, you have the factorization for the equation with the z's.
z6 – 26z3 – 27 = (z3 – 27)(z3 + 1) = 0
Then you then set each factor equal to zero. When z3 – 27 = 0, z3 = 27, and z = 3. And when z3 + 1 = 0, z3 = –1, and z = –1.
You can just take the cube roots of each side of the equations you form, because when you take that odd root, you know you can find only one real solution.
Here's another example. When solving the quadratic-like trinomial
y4 – 17y2 + 16 = 0,
you can factor the left side and then factor the factors again:
y4 – 17y2 + 16 = (y2 – 16)(y2 – 1) = (y – 4)(y + 4)(y – 1)(y + 1) = 0.
Setting the individual factors equal to zero, you get y = 4, y = –4, y = 1, y = –1.
