If 1 and ‒2 are two zeros of the polynomial (x³ ‒ 4x² ‒ 7x + 10), find its third zero.

Question

If 1 and ‒2 are two zeros of the polynomial (x 3 ‒ 4x 2 ‒ 7x + 10), find its third zero.

Philoid · Accepted Answer

Let us assume f (x) = x³ – 4x² – 7x + 10

As 1 and – 2 are the zeros of the given polynomial therefore each one of (x – 1) and (x + 2) is a factor of f (x)

Consequently, (x – 1) (x + 2) = (x² + x – 2) is a factor of f (x)

Now, on dividing f (x) by (x² + x – 2) we get:

f (x) = 0

(x² + x – 2) (x – 5) = 0

(x – 1) (x + 2) (x – 5) = 0

∴ x = 1 or x = - 2 or x = 5

Hence, the third zero is 5

If 1 and ‒2 are two zeros of the polynomial (x3 ‒ 4x2 ‒ 7x + 10), find its third zero.