using substitution solving polynomial equations

Question

Using Substitution Solving Polynomial Equations ?

Solve : (x³ + 4) 2 - 15 (x³ + 4) + 36 = 0.

You might be tempted to multiply everything out and factor. However, there's an easier way. Notice that this equation takes the form

A² - 15A + 36 = 0,

where A = x³ + 4. You can easily solve this simpler quadratic equation! The left side factors to give

(A - 12)(A - 3) = 0

so A = 12 or A = 3.

Since A = x³ + 4, this is the same as saying

x³ + 4 = 12 or x³ + 4 = 3,

So x³ = 8 or x³ = -1,

which means

x = 2 or x = -1.

Normally, you can do this substitution of A for x3 + 4 in your head. You can simply write:

(x³ + 4) 2 - 15(x³ + 4) + 36 = 0

((x³ + 4) - 12)((x³ + 4) - 3) = 0

x³ + 4 = 12 or x³ + 4 = 3,

and so on.