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Question

Verify using factor theorem, whether 2x46x3+3x2+3x2 is divisible by x23x+2 or not?

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Solution

The divisor is not a linear polynomial, it is a quadratic polynomial.
Let us factorize the divisor :
x23x+2
= x22xx+2
= x(x2)1(x2)
= (x1)(x2)
Then (x1) and (x2) are factors of polynomial x23x+2.

Let p(x)=2x46x3+3x2+3x2

If (x1) and (x2) both are factors of p(x), then for x=1 and x=2, p(x) should be 0.

On replacing x by 1, we get
p(1)=2(1)46(1)3+3(1)2+3(1)2
p(1)=26+3+32=0

On replacing x by 2, we get
p(2)=2(2)46(2)3+3(2)2+3(2)2
p(2)=3248+12+62=0

So, (x1) and (x2) are factors of p(x),
Hence, p(x)=2x46x3+3x2+3x2 is divisible by x23x+2.

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