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Check whether g(x) is a factor of p(x) by dividing polynomial p(x) by polynomial g(x), where p(x) = x5 − 4x3 + x2 + 3x + 1, g(x) = x3 − 3x + 1 - Mathematics

Sum

Check whether g(x) is a factor of p(x) by dividing polynomial p(x) by polynomial g(x),
where p(x) = x5 − 4x3 + x2 + 3x +1, g(x) = x3 − 3x + 1

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Solution

We have been given two polynomials

P(x) = x5 - 4x3 + x2 + 3x + 1 and g(x) = x3 - 3x + 1

We will say g(x) is factor of p(x) if remainder is zero when we divide p(x) by g(x).

x3 -3x + 1)`("x"^2-1)/("x"^5-4"x"^3+"x"^2+3"x"+1)`
                  `"x"^5-"x"^3+"x"^2`
                 -     +       -        
                -x3 + 3x +1 
                -x3 + 3x - 1                         
                +     -     +          
                                2       

Here, the remainder is 2 ≠ 0
g(x) is not a factor of p(x)

Notes

x3 -3x + 1)`("x"^2-1)/("x"^5-4"x"^3+"x"^2+3"x"+1)`
           `"x"^5-"x"^3+"x"^2`
           -     +       -         
          -x3 + 3x +1
          -x3 + 3x - 1 
          +     -      +          

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