Q. x – 4 and 2x – 1 are factors of which of the following polynomials?

Correct Answer: (B) 2x³ – 7x² – 5x + 4

Explanation:

To determine which polynomial has both x – 4 and 2x – 1 as factors, we can use the Factor Theorem. According to this theorem:

– If x – c is a factor of a polynomial P(x), then P(c) = 0.
– Similarly, if 2x – 1 is a factor, then P(1/2) = 0.

Option (A): 2x^3 – 7x^2 + 5x – 4

1. Check x = 4:
P(4) = 2(4)^3 – 7(4)^2 + 5(4) – 4 = 128 – 112 + 20 – 4 = 32 ≠ 0
Since P(4) ≠ 0, x – 4 is not a factor.

Option (B): 2x^3 – 7x^2 – 5x + 4

1. Check x = 4:
P(4) = 2(4)^3 – 7(4)^2 – 5(4) + 4 = 128 – 112 – 20 + 4 = 0
x – 4 is a factor.

2. Check x = 1/2:
P(1/2) = 2(1/2)^3 – 7(1/2)^2 – 5(1/2) + 4 = 1/4 – 7/4 – 5/2 + 4 = 0
2x – 1 is also a factor.

Option (C): 2x^3 + 7x^2 + 5x + 4

1. Check x = 4:
P(4) = 2(4)^3 + 7(4)^2 + 5(4) + 4 = 128 + 112 + 20 + 4 = 264 ≠ 0
x – 4 is not a factor.

Option (D): 2x^3 + 7x^2 – 5x – 4
1. Check x = 4:
P(4) = 2(4)^3 + 7(4)^2 – 5(4) – 4 = 128 + 112 – 20 – 4 = 216 ≠ 0
x – 4 is not a factor.

Conclusion: Only Option (B) satisfies both conditions P(4) = 0 and P(1/2) = 0.

Answer: (B) 2x³ – 7x² – 5x + 4