List the distinct roots of the following equation. For both repeated and single roots, specify their multiplicity. Enter $\left( r _ { 1 } , m _ { 1 } \right) , \left( r _ { 2 } , m _ { 2 } \right)$ , ... where $r _ { 1 } , r _ { 2 }$ etc. are the roots of the polynomial and $m _ { 1 }$ is the multiplicity of $r _ { 1 } , m _ { 2 }$ is the multiplicity of $r _ { 2 }$ etc. $( x - 3 ) ( x - 2 ) ^ { 6 } ( x - 9 ) = 0$
A) (3, 1), (2, 4), (9, 1)
B) (3, 1), (2, 1), (9, 1)
C) (3, 1), (2, 5), (9, 1)
D) (3, 1), (2, 6), (9, 1)
E) (3, 4), (2, 4), (9, 4)

Question

Quizplus · Accepted Answer

By the fundamental theorem of algebra, we know that a polynomial of degree n has exactly n roots (counting multiplicities). Therefore, we need to factor the given polynomial to determine its roots and their multiplicities.
We can distribute and simplify the given polynomial as follows: 
$$(x-3)(x-2)^6(x-9) = (x-3)(x-2)(x-2)(x-2)(x-2)(x-2)(x-2)(x-2)(x-9)$$
From this, we can see that the roots and their multiplicities are:
$$\left(3,1\right), \left(2,6\right), \left(9,1\right)$$
Therefore, the answer is choice D.

List the Distinct Roots of the Following Equation (r1,m1),(r2,m2)\left( r _ { 1 } , m _ { 1 } \right) , \left( r _ { 2 } , m _ { 2 } \right)(r1​,m1​),(r2​,m2​)

List the Distinct Roots of the Following Equation $\left( r _ { 1 } , m _ { 1 } \right) , \left( r _ { 2 } , m _ { 2 } \right)$