if -1 is one of the zeroes of py 3x^3-5x^2-11x-3 find other zeroes

Given:

The function is:

[tex]y=3x^3-5x^2-11x-3[/tex]

It is given that -1 is a zero of the given function.

To find:

The other zeroes of the given function.

Solution:

If c is a zero of a polynomial P(x), then (x-c) is a factor of the polynomial.

It is given that -1 is a zero of the given function. So, [tex](x+1)[/tex] is a factor of the given function.

We have,

[tex]y=3x^3-5x^2-11x-3[/tex]

Split the middle terms in such a way so that we get (x+1) as a factor.

[tex]y=3x^3+3x^2-8x^2-8x-3x-3[/tex]

[tex]y=3x^2(x+1)-8x(x+1)-3(x+1)[/tex]

[tex]y=(x+1)(3x^2-8x-3)[/tex]

Again splitting the middle term, we get

[tex]y=(x+1)(3x^2-9x+x-3)[/tex]

[tex]y=(x+1)(3x(x-3)+1(x-3))[/tex]

[tex]y=(x+1)(3x+1)(x-3)[/tex]

For zeroes, [tex]y=0[/tex].

[tex](x+1)(3x+1)(x-3)=0[/tex]

[tex](x+1)=0[/tex] and [tex]3x+1=0[/tex] and [tex]x-3=0[/tex]

[tex]x=-1[/tex] and [tex]x=-\dfrac{1}{3}[/tex] and [tex]x=3[/tex]

Therefore, the other two zeroes of the given function are [tex]-\dfrac{1}{3}[/tex] and [tex]3[/tex].

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