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Question
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
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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