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Question

What first degree polynomial added to 3x32x2 gives a polynomial for which both x − 1 and x + 1 are factors?

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Solution

Let the first degree polynomial that has to be added to the polynomial 3x3 2x2 be ax + b.

So, the polynomial is 3x3 2x2 + ax + b.

Divisor = (x 1)

Factor theorem says that for the polynomial p(x) and for the number a, if we have p(a) = 0, then (x − a) is a factor of p(x).

Thus, we must have:

3(1)3 2(1)2 + a(1) + b = 0 for (x − 1) to be a factor of 3x3 2x2 + ax + b.

3 2 + a + b = 0

a + b = 1 …(1)

Similarly, checking for (x + 1).

(x + 1) = {x − (1)}

We must have:

3(1)3 2(1)2 + a(1) + b = 0 for (x + 1) to be a factor of 3x3 2x2 + ax + b.

⇒ −3 2 a + b = 0

b a = 5 …(2)

Adding (1) and (2):

2b = 4

b = 2

a = 3

Therefore, the first degree expression that has to be added to 3x3 2x2 is 3x + 2, so that the polynomial obtained has both (x + 1) and (x 1) as the factors.


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