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.
![f\left( x \right) = a{x^n} + b{x^{n - 1}} + \ldots + cx + d](/tpl/images/0314/2178/c094e.png)
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}.](/tpl/images/0314/2178/dea6a.png)
Explanation:
The given equation is ![f\left( x \right) = {\left( {x - 6} \right)^2} \times {\left( {x + 2} \right)^2}.](/tpl/images/0314/2178/dea6a.png)
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}](/tpl/images/0314/2178/6051b.png)
The roots are ![- 2{\text{ and 6 with multiplicity 2}}](/tpl/images/0314/2178/f5bf0.png)
The roots of
and
.Option (d) and option (f) are correct.
Learn more:
Learn more about inverse of the functionhttps://link.
Learn more about equation of circle link.
Learn more about range and domain of the function link.
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.