To factor the expression \( x^4 - 10x^2 + 16 \), we can start by making a substitution. Let \( y = x^2 \). Then the expression becomes:
\[ y^2 - 10y + 16 \]
Next, we need to factor the quadratic \( y^2 - 10y + 16 \). We are looking for two numbers that multiply to \( 16 \) and add to \( -10 \). The numbers that meet these criteria are \( -2 \) and \( -8 \). Therefore, we can factor the quadratic as:
\[ (y - 2)(y - 8) \]
Now, we substitute back \( y = x^2 \):
\[ (x^2 - 2)(x^2 - 8) \]
Next, we can factor \( x^2 - 8 \), which can be written as \( x^2 - 4 \cdot 2 \). This can be factored further using the difference of squares:
\[ x^2 - 8 = x^2 - (2\sqrt{2})^2 = (x - 2\sqrt{2})(x + 2\sqrt{2}) \]
Thus we have:
\[ x^2 - 2 \quad \text{(as is)} \quad \text{and} \quad (x - 2\sqrt{2})(x + 2\sqrt{2}) \]
So the completely factored form of the original expression is:
\[ (x^2 - 2)(x - 2\sqrt{2})(x + 2\sqrt{2}) \]
In summary, the complete factorization of \( x^4 - 10x^2 + 16 \) is:
\[ (x^2 - 2)(x - 2\sqrt{2})(x + 2\sqrt{2}) \]
This is the factored form of the given expression.
1 answer