Use factoring and the root method to solve the polynomial equation.

Question

x Superscript 4 Baseline minus 11 x squared plus 10equals0
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Part 1
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​(Simplify your answer. Type exact​ answers, using radicals as needed. Use a comma to separate answers as​ needed.)

Answer 1

To solve the polynomial equation \( x^4 - 11x^2 + 10 = 0 \) using factoring and the root method, we can start by performing a substitution. Let \( y = x^2 \). This transforms the original equation into a quadratic equation in terms of \( y \):

\[ y^2 - 11y + 10 = 0 \]

Next, we can factor the quadratic:

\[ (y - 1)(y - 10) = 0 \]

Setting each factor equal to zero gives us:

\[ y - 1 = 0 \quad \text{or} \quad y - 10 = 0 \]

This results in:

\[ y = 1 \quad \text{or} \quad y = 10 \]

Now we will substitute back for \( x \):

1. For \( y = 1 \): \[ x^2 = 1 \] Taking the square root of both sides, we get: \[ x = \pm 1 \]

2. For \( y = 10 \): \[ x^2 = 10 \] Taking the square root of both sides, we get: \[ x = \pm \sqrt{10} \]

Combining all the solutions, we have:

\[ x = 1, -1, \sqrt{10}, -\sqrt{10} \]

Thus, the complete solution set is:

\[ \boxed{1, -1, \sqrt{10}, -\sqrt{10}} \]