Question

Find the value of α and β if the division of x^2010 + αx^2011 +β by (x – 1) leaves the remainder zero, (x + 1) leaves the remainder 4.

Answer 1

Step-by-step explanation:

Since the quadratic polynomial is divisible by both x+1 and x+2, it will be of the form k(x+1)(x+2),k being a constant.

Thus, it looks like kx

2

+3kx+2k.

Now, when divided by x+3, it leaves remainder 4.

So, kx

2

+3kx+2k−4 is exactly divisible by x+3.

Dividing the two, first term in the quotient will be kx.

Now, kx(x+3) is kx

2

+3kx, which when subtracted from the dividend gives 2k−4.

Since the remaining term is just a constant, it has to be zero for the polynomial to exactly divide it.

Hence, 2k−4=0, implying k to be 2.

And the required polynomial to be 2x

2

+6x+4.

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