ANSWER

The correct options are ,

D. -6 with multiplicity of 2

and

F. -2 with multiplicity of 2

EXPLANATION

The multiplicity of the roots of a polynomial function tells us the number of times the roots the roots of the polynomial repeat.

Considering the polynomial function

the roots can be found by equating either factors to zero

The square tells us that the root

repeats twice. This means it has a multiplicity of 2.

Also

tells us that the root

has a multiplicity of 2

Therefore options D and F are the correct options.

The correct options are ,

D. -6 with multiplicity of 2

and

F. -2 with multiplicity of 2

EXPLANATION

The multiplicity of the roots of a polynomial function tells us the number of times the roots the roots of the polynomial repeat.

Considering the polynomial function

the roots can be found by equating either factors to zero

The square tells us that the root

repeats twice. This means it has a multiplicity of 2.

Also

tells us that the root

has a multiplicity of 2

Therefore options D and F are the correct options.

theres only the two answers 6 with multiplicity 2 and -2 with multiplicity 2

Step-by-step explanation:

The roots of are and . Option (d) and option (f) are correct.

Further explanation:

The Fundamental Theorem of Algebra states that the polynomial has n roots if the degree of the polynomial is n.

The polynomial function has n roots or zeroes.

Given:

The equation is

Explanation:

The given equation is

According to the Fundamental Theorem of Algebra the function has 4 roots.

Solve the above equation to obtain the zeros.

The roots are

The roots of and .Option (d) and option (f) are correct.

Learn more:

Learn more about inverse of the Learn more about equation of circle Learn more about range and domain of the functionAnswer details:

Grade: High School

Subject: Mathematics

Chapter: Polynomial

Keywords: roots, linear equation, quadratic equation, zeros, function, polynomial, solution, cubic function, degree of the function, multiplicity of 1, multiplicity of 2.

6 with multiplicity 2

-2 with multiplicity 2

Step-by-step explanation:

We have given that find the roots of:

f (x) = (x- 6)²(x + 2)²

Note that the polynomials are already in the factored form. So we will only make it equal to zero

(x-6)² = 0

The square tells us that the root repeat twice

Move -6 to the R.H.S

Then,

x=6 , 6

(x+2)²=0

Move 2 to the R.H.S

Then,

x= -2, -2

Therefore the correct options are:

6 with multiplicity 2

-2 with multiplicity 2

X=6 , X = -2

Step-by-step explanation:

For any polynomial given in factorised form , the roots are determined as explained below:

Suppose we have a polynomial

(x-m)(x-n)(x+p)(x-q)

The roots of above polynomial will be m,n,-p, and q

Also if we have same factors more than once , there will be duplicate roots.

Example

(x-m)^2(x-n)(x+p)^2

For above polynomial

There will be total 5 roots . Out of which 2(m and -p) of them will be repeated. x=m , x=n , x =-p

Hence in our problem

The roots of f(x)= (x-6)^2(x+2)^2 are

x=6 And x=-2

..

Assumption: is defined for all (all real values of .)

Step-by-step explanation:

Evaluating for a root of this function shall give zero.

Equate and zero to find the root(s) of .

.

.

Multiply both sides by 1/4:

.

.

This polynomial has two factors:

(x - 6), and(x + 2) = (x - (-2)).Apply the factor theorem:

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