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Question

Use Remainder theorem to factorize the following polynomial. 2x3+3x29x10.

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Solution

Let f(x)=2x3+3x29x10, then f(1)=2+3+910=0
So, by remainder theorem, when f(x) is divided by x+1, the remainder is 0. So, (x+1) is a factor of f(x)
f(2)=16+121810=0
So, by similar reasoning, (x2) is also a factor of f(x)
Therefore, f(x)=(x+1)(x2)p(x)=(x2x2)p(x)
By synthetic division, we can find p(x)
f(x)=2x3+3x29x10=(x2x2)(2x+5)
Therefore p(x)=2x+5 and the factorization of f(x) is
f(x)=(x+1)(x2)(2x+5)

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