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To factor the polynomial [tex]\(x^2 - 62\)[/tex], you need to follow these steps:
1. Recognize the polynomial structure: The given polynomial is [tex]\(x^2 - 62\)[/tex], which is a difference of squares.
2. Recall the formula for difference of squares: The difference of squares formula is [tex]\(a^2 - b^2 = (a - b)(a + b)\)[/tex]. Here, [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are any expressions.
3. Identify the expressions for [tex]\(a\)[/tex] and [tex]\(b\)[/tex]: In the polynomial [tex]\(x^2 - 62\)[/tex], we can see it as: [tex]\[
a^2 = x^2 \quad \text{and} \quad b^2 = 62
\][/tex] Therefore, [tex]\[
a = x \quad \text{and} \quad b = \sqrt{62}
\][/tex]
4. Apply the difference of squares formula: Substitute [tex]\(a\)[/tex] and [tex]\(b\)[/tex] into the formula: [tex]\[
x^2 - 62 = (x - \sqrt{62})(x + \sqrt{62})
\][/tex]
So, the factors of the polynomial [tex]\(x^2 - 62\)[/tex] are: [tex]\[
(x - \sqrt{62})(x + \sqrt{62})
\][/tex]
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