4x³+12x²-x-3 is divisible by

Correct Answer is : Both (2x + 1) and (2x - 1)

Correct Answer is : Both (2x + 1) and (2x - 1)

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UPSC CDS 1 2021 Math Paper

Question : 75 of 100

Marks: +1, -0

4x^{3}+12x^{2}-x-3

is divisible by

Solution:

Concept:

Factorising the polynomial by grouping and using difference of squares.

Explanation:

Let

P(x)=4x^{3}+12x^{2}-x-3

Group terms:

(4x^{3}+12x^{2})+(-x-3)

Factor out common factors:

4x^{2}(x+3)-1(x+3)

Now

P(x)=(4x^{2}-1)(x+3)

4x^{2}-1

is a difference of squares:

(2x)^{2}-1^{2}=(2x-1)(2x+1)

Thus

P(x)=(2x-1)(2x+1)(x+3)

Both

(2x-1)

and

(2x+1)

are factors of

P(x)

Therefore

4x^{3}+12x^{2}-x-3

is divisible by each of these factors.

Answer:

Option C – Both

(2x+1)

and

(2x-1)

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