Ques 8:
The Volume(V) in cubic feet of an aquarium id modeled by the polynomial function V(x)= [tex] x^{3}+2x^{2}-13x+10 [/tex]
a) We have to explain that why x =4 is not a possible rational zero.
By Factor theorem, which states that a polynomial f(x) has a factor (x - k) if and only if f(k)=0.
For this , we will substitute the value of x in the given function.
[tex] V(x)=x^{3}+2x^{2}-13x+10 [/tex]
[tex] V(4)=4^{3}+2(4)^{2}-13(4)+10 [/tex]
[tex] V(4)=4^{3}+2(4)^{2}-13(4)+10 [/tex]
[tex] V(4)=54 [/tex] which is not equal to zero.
Therefore, x=4 is not a possible rational zero.
(b) To show that (x-1) is a factor of V(x).
By Factor theorem, which states that a polynomial f(x) has a factor (x - k) if and only if f(k)=0.
Let (x-1)=0
So, x=1.
Substituting x=1 in the given function.
[tex] V(1)=1^{3}+2(1)^{2}-13(1)+10 [/tex]
[tex] V(1)= -10+10 [/tex]
V(1) = 0
Therefore, (x-1) is a factor of V(x).
Now we will factorize the given function.
Dividing the given function by (x-1).
On dividing, we get quotient as [tex] x^{2}+3x-10 [/tex]
So, factored form is = [tex] (x-1)(x^{2}+3x-10) [/tex]
= [tex] (x-1)(x^{2}+5x-2x-10) [/tex]
= [tex] (x-1)(x(x+5)-2(x+5)) [/tex]
=[tex] (x-1)(x+5)(x-2) [/tex]
(c) So, the dimensions are 1,2 and -5.
