Question

Which polynomial function f(x) has a leading coefficient of 1, roots –4, 2, and 9 with multiplicity 1, and root –5 with multiplicity 3?

f(x) = 3(x + 5)(x + 4)(x – 2)(x – 9)
f(x) = 3(x – 5)(x – 4)(x + 2)(x + 9)
f(x) = (x + 5)(x + 5)(x + 5)(x + 4)(x – 2)(x – 9)
f(x) = (x – 5)(x – 5)(x – 5)(x – 4)(x + 2)(x + 9)

Answer 1

"polynomial function f(x) has a leading coefficient of 1"

So because of that, we can eliminate A and B.

Now we have "root –5 with multiplicity 3"

That's just product of three factors (x+5)

So D is wrong.

We are left with the only possible correct option C.

Answer 2

The expression for f(x) is:

f(x) = (x+5)(x+5)(x+5)(x+4)(x-2)(x-9)

Explanation:

We know that for any polynomial equation with roots:

`a_1,a_2,a_3,...` with multiplicity:

`m_1,m_2,...`

the equation for the polynomial is given by:

`f(x)=(x-a_1)^(m_1)(x-a_2)^(m_2)......`

if the leading coefficient is negative we may take '-' sign in the starting of the expression.

Here we are given that :

f(x) has a leading coefficient of 1, roots –4, 2, and 9 with multiplicity 1, and root –5 with multiplicity 3

Hence, f(x) is given by:

`f(x)=(x-(-4))^(1)(x-2)^(1)(x-9)^(1)(x-(-5))^(3)\\\\\\i.e.\\\\\\f(x)=(x+4)(x-2)(x-9)(x+5)^3\\\\\\f(x)=(x+5)(x+5)(x+5)(x+4)(x-2)(x-9)`

Hence, the expression for f(x) is:

`f(x)=(x+5)(x+5)(x+5)(x+4)(x-2)(x-9)`