Question

If two zeroes of a polynomial x3+5x2+7x+3 are -1 and-3 then find the third Zero

Answer 1

Third zero is -1.

Explanation:

consider the given polynomial is

`P(x)=x^3+5x^2+7x+3`

It is given that -1 and -3 are two zeroes, therefore (x+1) and (x+3) are two factors of given polynomial.

The given polynomial can be rewritten as

`P(x)=x^3+x^2+4x^2+4x+3x+3`

`P(x)=x^2(x+1)+4x(x+1)+3(x+1)`

`P(x)=(x+1)(x^2+4x+3)`

Now, splitting the middle term, we get

`P(x)=(x+1)(x^2+3x+x+3)`

`P(x)=(x+1)(x(x+3)+1(x+3))`

`P(x)=(x+1)(x+1)(x+3)`

`P(x)=(x+1)^2(x+3)`

Here, power of (x+1) is 2. It means the multiplicity of zero -1 is 2.

So, three zeroes are -1, -1 and -3.

Therefore the third zero is -1.