Which of the following describes the zeroes of the graph of f(x) = 3x6 + 30x5 + 75x4?

ANSWER

The zeros are:
[tex]x = 0 \: \: or \: \: x = - 5[/tex]
with multiplicity of 4 and 2 respectively.


EXPLANATION

The given function is

[tex]f(x) =3 {x}^{6} + 30 {x}^{5} + 75 {x}^{4} [/tex]
We want to find

[tex]3 {x}^{6} + 30 {x}^{5} + 75 {x}^{4} = 0 [/tex]

We can find the zeros of this function by factorizing the greatest common factor to obtain,

[tex]3 {x}^{4} ( {x}^{2} + 10x + 25) = 0[/tex]

The expression in the bracket can be rewritten as,

[tex]3 {x}^{4} ( {x}^{2} + 2(5)x + {5}^{2} ) = 0[/tex]

We can see clearly that, the expression in the bracket is a perfect square that can be factored as,

[tex]3 {x}^{4} ( x+ 5)^2 = 0[/tex]

This implies that,

[tex]3 {x}^{4} = 0[/tex]
or

[tex](x + 5) ^{2} = 0[/tex]

This gives,

[tex] {x}^{4} = 0 \: \: or \: \: x + 5 = 0[/tex]

This finally gives,

[tex]x = 0 \: \: or \: \: x = - 5[/tex]