Question

Use factoring and the root method to solve the polynomial equation.
x Superscript 4 Baseline minus 3 x squared plus 2equals0
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Part 1
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Answer 1

To solve the polynomial equation \( x^4 - 3x^2 + 2 = 0 \) using factoring, we can first make a substitution to simplify the expression. Let's set \( y = x^2 \). This gives us:

\[
y^2 - 3y + 2 = 0
\]

Next, we can factor this quadratic equation. We look for two numbers that multiply to \( 2 \) (the constant term) and add to \( -3 \) (the coefficient of the \( y \) term). The numbers \( -1 \) and \( -2 \) fit these criteria:

\[
(y - 1)(y - 2) = 0
\]

Now, we can solve for \( y \):

1. \( y - 1 = 0 \) leads to \( y = 1 \)
2. \( y - 2 = 0 \) leads to \( y = 2 \)

Recalling that \( y = x^2 \), we substitute back to find values for \( x \):

1. For \( y = 1 \):
\[
x^2 = 1 \implies x = \pm 1
\]

2. For \( y = 2 \):
\[
x^2 = 2 \implies x = \pm \sqrt{2}
\]

Putting it all together, the solutions for the original polynomial equation \( x^4 - 3x^2 + 2 = 0 \) are:

\[
x = 1, \, x = -1, \, x = \sqrt{2}, \, x = -\sqrt{2}
\]

So the final answer is:

\[
x = 1, -1, \sqrt{2}, -\sqrt{2}
\]