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Answer 1

Ques 8:

The Volume(V) in cubic feet of an aquarium id modeled by the polynomial function V(x)=
`x^(3)+2x^(2)-13x+10`

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.

`V(x)=x^(3)+2x^(2)-13x+10`

`V(4)=4^(3)+2(4)^(2)-13(4)+10`

`V(4)=4^(3)+2(4)^(2)-13(4)+10`

`V(4)=54` 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.

`V(1)=1^(3)+2(1)^(2)-13(1)+10`

`V(1)= -10+10`

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
`x^(2)+3x-10`

So, factored form is =
`(x-1)(x^(2)+3x-10)`

=
`(x-1)(x^(2)+5x-2x-10)`

=
`(x-1)(x(x+5)-2(x+5))`

=
`(x-1)(x+5)(x-2)`

(c) So, the dimensions are 1,2 and -5.