Question

What are the zeros of the polynomial function f(x) = x3 - 10x2 + 24x? I just want to know the name of the process to find the zeros.

Answer 1

f(x) = x³- 10x² + 24x

f(x) = x ( x² + 10x + 24)

f(x) = x (x + 6)(x+4)

x=0


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x+6 = 0
x = 0-6
x= -6


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x+4 = 0
x = 0-4
x= -4

zero's would be: 0, -6 and -4,
So it is your third choice down.

Answer 2

A zero of a function is a number, when plugged in for the variable, makes the function equal to zero.

Then, the roots of a polynomial P(x) are values of x such that P(x) = 0.

Given the polynomial function:
`f(x)=x^3-10x^2+24x`

By the rational theorem process, gives us the following possible roots: 0,
`\pm 1`,
`\pm 2`,
`\pm 3` ,
`\pm 4`,
`\pm 6`,
`\pm 8`,
`\pm 12` and
`\pm 24`

for x =0

`f(0)=0^3-10(0)^2+24(0)=0`

Now, our polynomial become:

`x(x^2-10x+24)` = 0

Then, we factors the remaining quadratic equation, factoring by grouping , using the facts 4+6 = 10 and
`4 \cdot 6 = 24`

`x(x^2-6x-4x+24)` = 0

`x(x(x-6)-4(x-6))` =0

`x((x-6)(x-4))` =0

Zero product property states that if xy = 0 then either a =0 or b =0.

by zero product property;

⇒ x = 0, x-6=0 and x-4 = 0

Hence, x = 0 , x = 4 and x =6 are the zeros of the given polynomial function.