Question

Please help, need answer fast!

Which of the following describes the roots of the polynomial function f(x)= (x + 2)^2(x - 4)(x + 1)^3?

–2 with multiplicity 2, 4 with multiplicity 1, and –1 with multiplicity 3
–2 with multiplicity 3, 4 with multiplicity 2, and –1 with multiplicity 4
2 with multiplicity 2, –4 with multiplicity 1, and 1 with multiplicity 3
2 with multiplicity 3, –4 with multiplicity 2, and 1 with multiplicity 4

Answer 1

Multiplicity means multiple roots. So
`(x + a)^n` means that the root
`-a` has multiplicity
`n`.

Using the definition of multiplicity of roots, we deduce that we have:

(A) -2 with multiplicity 2, 4 with multiplicity 1, and -1 with multiplicity 3.

Answer 2

Multiplicity of a polynomial means how many times a particular number is a zero for a given polynomial.

In the given polynomial :

`f(x)=(x+2)^(2) (x-4)(x+1)^(3)`

The roots of the equation can be found by taking the factor =0.

x+2=0 or x=-2

x-4=0

or x=4

x+1=0

or x=-1

The roots of the polynomial are -2,4,-1.

The powers of the root denotes the multiplicity of the polynomial.

The root -2 occurs 2 times ,4 occurs once ,-1 occurs 3 times.

So we say :–2 with multiplicity 2, 4 with multiplicity 1, and –1 with multiplicity 3.

Option A is the right option.