Question

question 3
find the partition number(s) of ( r(x) = \frac{x^2(x^2 - 4)}{(x + 3)(16)} )
choose all that apply.
□ ( x = -3 )
□ ( x = -4 )
□ ( x = -2 )
□ ( x = 3 )
□ ( x = 4 )
□ ( x = 0 )
□ ( x = 16 )
□ ( x = 2 )

Explanation:

Step1: Define partition numbers

Partition numbers of a rational function are x-values where the function is undefined or its numerator equals 0.

Step2: Find undefined x-values

Set denominator equal to 0:
$(x+3)(16)=0$
Solve for x: $x+3=0 \implies x=-3$

Step3: Find numerator roots

Set numerator equal to 0:
$x^2(x^2-4)=0$
Factor $x^2-4$: $x^2(x-2)(x+2)=0$
Solve for x: $x=0$, $x=2$, $x=-2$

Step4: Compile all partition numbers

Combine values from Step2 and Step3: $x=-3, x=-2, x=0, x=2$

Answer:

A. x=-3, C. x=-2, F. x=0, H. x=2