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1.Determine which of the following polynomials has a factor :
i)\(x^3 + x^2 + x + 1\)
ii)\(x^4 + x^3 + x^2 + x + 1\)
iii)\(x^4 + 3x^3 + 3x^2 + x + 1\)
iv)\(x^3 - x^2 - (2 + \sqrt{2})x + \sqrt{2}\)
Answer :

The zero of x + 1 is –1.
i)Let \(p(x) = x^3 + x^2 + x + 1\)
So, To check, whether x + 1 is a factor of \(x^3 + x^2 + x + 1\).
By factor theorem,
\(p(–1) = {–1}^3 + {–1}^2 + {-1} + 1\)
i.e. \(p(–1) = –1 + 1 -1 + 1\)
i.e.\(p(-1) = 0\)
Hence, x + 1 is the factor of \(x^3 + x^2 + x + 1\)

ii)Let \(p(x) = x^4 + x^3 + x^2 + x + 1\)
So, To check, whether x + 1 is a factor of \(x^4 + x^3 + x^2 + x + 1\).
By factor theorem,
\(p(–1) = {-1}^4 + {–1}^3 + {–1}^2 + {-1} + 1\)
i.e. \(p(–1) = 1 – 1 + 1 - 1 + 1\)
i.e.\(p(-1) = 1\)
Hence, x + 1 is not the factor of \(x^4 + x^3 + x^2 + x + 1\)

iii)Let \(p(x) = x^4 + 3x^3 + 3x^2 + x + 1\)
So, To check, whether x + 1 is a factor of \(x^4 + 3x^3 + 3x^2 + x + 1\).
By factor theorem,
\(p(–1) = {-1}^4 + 3{–1}^3 + 3{–1}^2 + {-1} + 1\)
i.e. \(p(–1) = 1 – 3 + 3 - 1 + 1\)
i.e.\(p(-1) = 1\)
Hence, x + 1 is not the factor of \(x^4 + 3x^3 + 3x^2 + x + 1\)

iv)Let \(p(x) = x^3 - x^2 - (2 + \sqrt{2})x + \sqrt{2}\)
So, To check, whether x + 1 is a factor of \(x^3 - x^2 - (2 + \sqrt{2})x + \sqrt{2}\).
By factor theorem,
\(p(–1) = {–1}^3 - {–1}^2 - (2 + \sqrt{2}){-1} + \sqrt{2}\)
i.e. \(p(–1) = – 1 - 1 + 2 + \sqrt{2} + \sqrt{2}\)
i.e.\(p(-1) = 2\sqrt{2}\)
Hence, x + 1 is not the factor of \(x^3 - x^2 - (2 + \sqrt{2})x + \sqrt{2}\)