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Which of the subsequent are the correct factors for the polynomial $2 x 3 − 3 x 2 − 17 x + 30$ ?

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x − 2, x + 3 and 2 x + 5

x + 2, x + 3 and 2 x − 5

x − 2, x − 3 and 2 x − 5

x − 2, x + 3 and 2 x − 5

answer is D.

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Detailed Solution

Let the given polynomial is,

p (x) = 2 x 3 − 3 x 2 − 17 x + 30

First, determine the prime factor of the given polynomial's constant term.
Find the minimum factor that satisfy

p x,

replace

x

with this factor, and then express the result in the form of a linear polynomial.
In the given polynomial

p x,

constant term = 30.
Therefore, the factors of 30 are,

± 1, ± 2, ± 3, ± 5, ± 6, ± 10, ± 15, ± 30

∵ p (2) = 223 − 322 − 172 + 30 ⇒ p (2) = 16 − 12 − 34 + 30 ⇒ p (2) = 0

By trial, we find that

p 2 = 0.

This shows that

x − 2

is a factor of

p x .

∴ 2 x 3 − 3 x 2 − 17 x + 30 = 2 x 3 − 4 x 2 + x 2 − 2 x − 15 x + 30 ⇒ 2 x 3 − 3 x 2 − 17 x + 30 = 2 x 2 x − 2 + x x − 2 − 15 x − 2 ⇒ 2 x 3 − 3 x 2 − 17 x + 30 = x − 2 2 x 2 + x − 15

Now, we use the concept of mid-term splitting to solve the quadratic polynomial

2 x 2 + x − 15 .

∴ 2 x 2 + x − 15 = 2 x 2 + 6 x − 5 x − 15 ⇒ 2 x 2 + x − 15 = 2 x x + 3 − 5 x + 3 ⇒ 2 x 2 + x − 15 = x + 3 2 x − 5

Therefore,

2 x 3 − 3 x 2 − 17 x + 30 = x − 2 x + 3 2 x − 5 .

Thus, the correct factors for the polynomial

2 x 3 − 3 x 2 − 17 x + 30

are

x − 2, x + 3 and 2 x − 5 .

Hence, option (4) is correct.

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