Answer:
Third zero is -1.
Step-by-step explanation:
consider the given polynomial is
[tex]P(x)=x^3+5x^2+7x+3[/tex]
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
[tex]P(x)=x^3+x^2+4x^2+4x+3x+3[/tex]
[tex]P(x)=x^2(x+1)+4x(x+1)+3(x+1)[/tex]
[tex]P(x)=(x+1)(x^2+4x+3)[/tex]
Now, splitting the middle term, we get
[tex]P(x)=(x+1)(x^2+3x+x+3)[/tex]
[tex]P(x)=(x+1)(x(x+3)+1(x+3))[/tex]
[tex]P(x)=(x+1)(x+1)(x+3)[/tex]
[tex]P(x)=(x+1)^2(x+3)[/tex]
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.
