Question

If 1 and -2 are the zeros of polynomial (x³-4x²-7x+10) find its third zero?​

Answer 1

Answer:

The zeroes of the polynomial are 1, -2, and 5 and the 3rd zero is 5

Step-by-step explanation:

We are given that,

1 and -2 are the zeroes of x³ - 4x² - 7x + 10

So,

x = 1 and x = -2

then,

(x - 1) and (x + 2) are factors of x³ - 4x² - 7x + 10

Now,

Let the other factor be y

So,

x³ - 4x² - 7x + 10 = (x - 1) × (x + 2) × y

because it is 3rd degree polynomial, so it will have 3 factors.

So,

y = x³ - 4x² - 7x + 10 ÷ ((x - 1) × (x + 2))

First we must find the product of the factors, then only we can find their Quotient,

So,

(x - 1)(x + 2)

Using Distributive property,

= x × x + 2 × x - 1 × x + - 1 × 2

= x² + 2x - x - 2

= x² + x - 2

Now,

to find y,

x - 5

_____________

x² + x - 2 | x³ - 4x² - 7x + 10

- (x³ + x² - 2x)

(-) (-) (+)

———————

0 - 5x² - 5x + 10

-(‐5x² - 5x + 10)

(+) (+) (-)

————————

0

————————

Thus,

x³ - 4x² - 7x + 10 = (x - 1) • (x + 2) • (x - 5)

Hence,

the 3rd factor is (x - 5)

So,

3rd zero will be

x = 5

Thus,

The zeroes of the polynomial are 1, -2, and 5 and the 3rd zero is 5

Hope it helped and believing you understood it........All the best.