Write the difference of squares pattern. Then, rewrite the binomial that has terms with opposite signs as a new difference of squares with new variables — one variable and one constant. Finally, rewrite the original binomial in terms of the new variables and solve it.
Example binomial: x^4-16 Solutions: x=-2 and x=2
Practice makes perfect
Let's begin by writing the difference of squares pattern.
a^2 - b^2 = ( a+ b)( a- b)
We are required to find a binomial where we need to apply this pattern twice to factor it. By looking at the expression above, it implies that the factor ( a- b) also has to be a difference of two squares. For example, it can be equal to x^2-2^2.
a- b = x^2-2^2
From the above equation we can consider a=x^2 and b=2^2. With these two new variables, the difference of squares looks as follows.
a^2- b^2 &= (x^2)^2-(2^2)^2
&= x^4-16
Next, we set our binomial equal to zero.
x^4-16 = 0
Finally, let's solve this equation.
Equation (I) has no real solutions. This means that the solutions to our binomial equation are x=-2 and x=2. Keep in mind that this is just an example binomial and your answer may vary.
Extra
Extra
Let's see why x^2+4=0 does not have real solutions.
