Q3 of 31 Page 245

By adding a number to p(x) = x3 + x2 + x, a new polynomial q(x) is to be formed.

What number should be added, so that x – 1 is a factor of q(x)?

Given p(x) = x3 + x2 + x


Let the number to be added be “k”.


Then, the new polynomial q(x) = x3 + x2 + x + k


Now, (x – 1) is a factor of x3 + x2 + x + k.


i.e. x = 1 is the root of the polynomial.


Then, put the polynomial to zero we get,


x3 + x2 + x + k = 0


⇒ (1)3 + (1)2 + 1 + k = 0


⇒ 1 + 1+1 + k = 0


⇒ k = – 3


Hence, “ – 3” should be added to the polynomial such that (x – 1) is a factor of q(x).


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By adding a number to p(x) = x3 + x2 + x, a new polynomial q(x) is to be formed.

What number should be added, so that x + 1 is a factor of q(x)?

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In each pair of polynomials below find what kind of natural number n must be, so that the first is a factor of the second.

x – 1, xn – 1