The polynomial p(x) = x^3 + x - 10. Show x-2 is a factor of p(x) and express p(x) in the form (x-2)(x^2+ax+b).

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justaguide | College Teacher | (Level 2) Distinguished Educator

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The polynomial p(x) = x^3 + x - 10. According to the factor theorem (x - 2) is a factor of p(x) if p(2) = 0

x^3 + x - 10

=> 8 + 2 - 10

=> 0

This proves that x - 2 is a factor of p(x).

Using the fact that x - 2 is a factor of p(x), it can be written as (x-2)(x^2+ax+b)

x^3 + x - 10

=> x^3 - 2x^2 + 2x^2 - 4x + 5x - 10

=> x^2(x - 2) + 2x(x - 2) + 5(x - 2)

=> (x - 2)(x^2 + 2x + 5)

The required factorized form of x^3 + x - 10 is (x - 2)(x^2 + 2x + 5)

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