Question

Find the root(s) of f (x) = (x- 6)2(x + 2)2.

Answer 1

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

$f(x) = {(x + 6)}^{2} {(x + 2)}^{2}$


the roots can be found by equating either factors to zero

${(x + 6)}^{2} = 0$

The square tells us that the root
$x = - 6$
repeats twice. This means it has a multiplicity of 2.


Also

$(x + 2) ^{2} = 0$
tells us that the root
$x = - 2$
has a multiplicity of 2


Therefore options D and F are the correct options.

Answer 2

The roots of $f\left( x \right) = {\left( {x - 6} \right)^2} \times {\left( {x + 2} \right)^2}$ are $\boxed{6{\text{ with multiplicity 2}}}$ and $\boxed{ - {\text{2 with multiplicity 2}}}$. 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.

$f\left( x \right) = a{x^n} + b{x^{n - 1}} + \ldots + cx + d$

The polynomial function has n roots or zeroes.

Given:

The equation is $f\left( x \right) = {\left( {x - 6} \right)^2} \times {\left( {x + 2} \right)^2}.$

Explanation:

The given equation is $f\left( x \right) = {\left( {x - 6} \right)^2} \times {\left( {x + 2} \right)^2}.$

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

Solve the above equation to obtain the zeros.

$\begin{aligned}f\left( x \right) &= {\left( {x - 6} \right)^2} \times {\left( {x + 2} \right)^2}\\0&= {\left( {x - 6} \right)^2} \times {\left( {x + 2} \right)^2}\\0&= {\left( {x - 6} \right)^2}{\text{ or }}{\left( {x + 2} \right)^2} &= 0\\0&= x - 6{\text{ or }}x + 2 &= 0\\6&= x{\text{ or }}x &= - 2 \\\end{aligned}$

The roots are $- 2{\text{ and 6 with multiplicity 2}}$

The roots of $f\left( x \right) = {\left( {x - 6} \right)^2} \times {\left( {x + 2} \right)^2}\[\text{are}\: \boxed{6{\text{ with multiplicity 2}}}$ and $\boxed{ - {\text{2 with multiplicity 2}}}$.Option (d) and option (f) are correct.

Learn more:

1. Learn more about inverse of the functionhttps://brainly.com/question/1632445.
2. Learn more about equation of circle brainly.com/question/1506955.
3. Learn more about range and domain of the function brainly.com/question/3412497.

Answer 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.