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Q.

The factors of the expression x³ – 2x² – x + 2 are:

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a

(x + 2) (x - 1) (x + 2)

b

(x + 2) (x - 2) (x + 2)

c

(x + 1) (x - 2) (x + 2)

d

(x + 1) (x - 1) (x + 2)

answer is D.

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Detailed Solution

Given: x³ – 2x² – x + 2
First, Let us find the factors of a constant term.
Secondly, let us check out at which factor the given polynomial becomes zero by using the hit and trial method and get one factor of the given polynomial. Then, we shall further divide the polynomial by this factor to get a quotient of the second degree.
Finally, we shall factorize the second-degree polynomial by middle term splitting to get required factors.
Let p(x) = x³ – 2x² – x + 2
Here constant term is 2.Now, let us find the factors of 2:
Factors of 2 are ±1 and ± 2.
By using hit and trial method substitute x = - 1 in p(x), we get

p (x) = x 3 − 2 x 2 − x + 2

p (− 1) = (− 1) 3 − 2 (− 1) 2 − (− 1) + 2

= − 1 − 2 + 1 + 2

= − 3 + 3

= 0

Therefore, (x + 1) is the factor of p(x).
By dividing p(x) by (x + 1), we get:
Question Image

Question Image

Then, p(x) = (x + 1) (x² – 3x + 2).
Here quotient is (x² – 3x + 2).
When we use the middle term splitting in the quotient, we get:

= (x + 1) (x 2 − x − 2 x + 2)

= (x + 1) x (x − 1) − 2 (x − 1)

= (x + 1) (x − 1) (x + 2)

Hence , the required factors of p(x) is (x + 1) (x - 1) (x + 2).
Therefore, the correct option is 4.

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