Question

Factor the expression completely.
x^4 - 10x^2 + 16

Answer 1

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.