Question

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

Answer 1

F(x) = 3x^6 + 30x^5 +75x^4
Factoring  3x^4(x^2 +10x + 25) = 3x^4(x+5)(x+5) = 0
x = 0, -5 with -5 having a multiplicity of 2 are the zeroes.

Answer 2

ANSWER

The zeros are:
$x = 0 \: \: or \: \: x = - 5$
with multiplicity of 4 and 2 respectively.


EXPLANATION

The given function is

$f(x) =3 {x}^{6} + 30 {x}^{5} + 75 {x}^{4}$
We want to find

$3 {x}^{6} + 30 {x}^{5} + 75 {x}^{4} = 0$

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

$3 {x}^{4} ( {x}^{2} + 10x + 25) = 0$

The expression in the bracket can be rewritten as,

$3 {x}^{4} ( {x}^{2} + 2(5)x + {5}^{2} ) = 0$

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

$3 {x}^{4} ( x+ 5)^2 = 0$

This implies that,

$3 {x}^{4} = 0$
or

$(x + 5) ^{2} = 0$

This gives,

${x}^{4} = 0 \: \: or \: \: x + 5 = 0$

This finally gives,

$x = 0 \: \: or \: \: x = - 5$