What are the factors of x3 + 4x2 – 11x – 30?

Detailed solution?

Getting the first root by trial and error method?

Since the sum of the coefficients is not zero, 1 is not a root

Let x = -2

⇒ -8 + 16 + 22 – 30 = 0

∴ -2 is or (x + 2) is a factor

Dividing x3 + 4x2 – 11x – 30 by (x + 2)

Quotient = x2 + 2x – 15

⇒ x2 + 5x – 3x – 15 = 0

⇒ x (x + 5) – 3 (x + 5) = 0

⇒ (x – 3) (x + 5) = 0

Other factors are (x – 3) and (x + 5)

∴ (x + 2), (x – 3) and (x + 5) are the roots of the equation

Alternative method?

Since all the given options have only 3, 5 and 2 in different signs, this shortcut can be used

Product of the roots = -d/a = 30, positive

Sum of the roots = -b/a = -4

For product to be positive, either all terms are positive or two terms are negative

But since sum is negative, two terms has to be negative

Trying all thee combination of two roots to be negative

2 – 5 – 3 = -6

3 – 5 – 2 = -4

Second combination matches the condition –b/a = 4

∴ The roots are 3, -5 and -2

∴ The factors are (x – 3), (x + 5) and (x + 2)