Answer:
See Explanation
Step-by-step explanation:
From your question, what you need is the explanation;
Given that [tex](x - 1 - \sqrt{2})(x - 1 + \sqrt{2})[/tex] is a factor of p(x)
Required: Simplify
[tex](x - 1 - \sqrt{2})(x - 1 + \sqrt{2})[/tex]
Group the expression above
[tex]((x - 1) - \sqrt{2})((x - 1) + \sqrt{2})[/tex] -------- Expression 1
The expression can be expanded using difference of two squares;
[tex](a - b)(a + b) = a^2 - b^2[/tex]
By comparison, Expression 1 can be rewritten as
[tex](x - 1)^2 - (\sqrt{2})^2[/tex]
Open both brackets
[tex](x - 1)(x - 1) - (\sqrt{2})(\sqrt{2})[/tex]
Expand the above expression
[tex](x(x - 1) -1(x - 1)) - 2[/tex]
[tex](x^2 - x -x + 1)- 2[/tex]
[tex](x^2 - 2x + 1)- 2[/tex]
Remove bracket
[tex]x^2 - 2x + 1- 2[/tex]
[tex]x^2 - 2x - 1[/tex]
Hence,
If
[tex](x - 1 - \sqrt{2})(x - 1 + \sqrt{2})[/tex] is a factor of p(x)
Then
[tex]x^2 - 2x - 1[/tex] is a factor of p(x)
