Question

select all the functions with x - intercepts at (-5,0), (1,0), and (6,0).
□ ( a(x) = (x - 5)(x + 1)(x + 6) )
□ ( b(x) = (x + 5)(x - 1)(x - 6) )
□ ( c(x) = (x - 5)^2(x - 1)^2(x - 6)^3 )
□ ( d(x) = -(x + 5)(x - 1)(x - 6) )
□ ( e(x) = (x + 5)^3(x - 1)^2(x - 6)^2 )

Explanation:

Step1: Recall x-intercept factor rule

For an x-intercept at $(a,0)$, the function has a factor of $(x-a)$.

Step2: Identify required factors

For intercepts $(-5,0)$, $(1,0)$, $(6,0)$, factors are $(x+5)$, $(x-1)$, $(x-6)$.

Step3: Check each function

• $a(x)=(x-5)(x+1)(x+6)$: Factors are $(x-5),(x+1),(x+6)$ → incorrect intercepts.
• $b(x)=(x+5)(x-1)(x-6)$: Matches required factors → correct.
• $c(x)=(x-5)^2(x-1)^2(x-6)^3$: Factor $(x-5)$ is wrong → incorrect intercepts.
• $d(x)=-(x+5)(x-1)(x-6)$: Matches required factors (negative sign doesn't change intercepts) → correct.
• $e(x)=(x+5)^3(x-1)^2(x-6)^7$: Matches required factors (exponents don't change intercepts) → correct.

Answer:

B. $b(x) = (x + 5)(x - 1)(x - 6)$
D. $d(x) = -(x + 5)(x - 1)(x - 6)$
E. $e(x) = (x + 5)^3(x - 1)^2(x - 6)^7$