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.
f(x) = x³- 10x² + 24x
f(x) = x ( x² + 10x + 24)
f(x) = x (x + 6)(x+4)
x=0
-----------------
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:
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.