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. Finally, rewrite the original binomial in terms of the new variables.
Example binomial: x^4-y^4. Factored form: (x^2+y^2)(x+y)(x-y).
Practice makes perfect
Let's begin by writing the difference of squares pattern.
a^2 - b^2 = ( a+ b)( a- b)
We want a binomial where we need to apply this pattern more than once to factor it completely. By looking at the expression above, it implies that the factor ( a- b) has to be also a difference of two squares.
a- b = x^2-y^2From above, we can consider a=x^2 and b=y^2. With these two new variables, the left-hand side expression looks as follows.
a^2- b^2 &= (x^2)^2-(y^2)^2
&= x^4-y^4
Next, let's factor it.
Let's write our binomial and its factored form.
x^4-y^4 = (x^2+y^2)(x+y)(x-y)
Keep in mind that this is just a sample binomial and your answer may vary.
Extra
Extra
Notice that we could repeat the process again and obtain a binomial that requires applying the difference of square pattern three times in order to factor it completely.
x^4-y^4 = (x^2+y^2)(x+y)(x-y)
To do that, we must consider the factor (x-y) to be a difference of squares with new variables.
x-y = p^2-q^2
From above, we can consider x=p^2 and y=q^2. Therefore, the binomial x^4-y^4 will change as follows.
x^4-y^4 &= (p^2)^4-(q^2)^2
&= p^8-q^8
Let's factor the final binomial.
