Find Where Undefined/Discontinuous (6k+30)/(33k^5)*(k^2+16k+55)/(2k^3+20k^2+50k)

$\frac{6 k + 30}{33 k^{5}} \cdot \frac{k^{2} + 16 k + 55}{2 k^{3} + 20 k^{2} + 50 k}$

Step 1

Set the denominator in $\frac{6 k + 30}{33 k^{5}}$ equal to $0$ to find where the expression is undefined.

$33 k^{5} = 0$

Step 2

Solve for $k$ .

Divide each term in $33 k^{5} = 0$ by $33$ and simplify.

Divide each term in $33 k^{5} = 0$ by $33$ .

$\frac{33 k^{5}}{33} = \frac{0}{33}$

Simplify the left side.

Cancel the common factor of $33$ .

Cancel the common factor.

$\frac{33 k^{5}}{33} = \frac{0}{33}$

Divide $k^{5}$ by $1$ .

$k^{5} = \frac{0}{33}$

Simplify the right side.

Divide $0$ by $33$ .

$k^{5} = 0$

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$k = \sqrt[5]{0}$

Simplify $\sqrt[5]{0}$ .

Rewrite $0$ as $0^{5}$ .

$k = \sqrt[5]{0^{5}}$

Pull terms out from under the radical, assuming real numbers.

$k = 0$

Step 3

Set the denominator in $\frac{k^{2} + 16 k + 55}{2 k^{3} + 20 k^{2} + 50 k}$ equal to $0$ to find where the expression is undefined.

$2 k^{3} + 20 k^{2} + 50 k = 0$

Step 4

Solve for $k$ .

Factor the left side of the equation.

Factor $2 k$ out of $2 k^{3} + 20 k^{2} + 50 k$ .

Factor $2 k$ out of $2 k^{3}$ .

$2 k (k^{2}) + 20 k^{2} + 50 k = 0$

Factor $2 k$ out of $20 k^{2}$ .

$2 k (k^{2}) + 2 k (10 k) + 50 k = 0$

Factor $2 k$ out of $50 k$ .

$2 k (k^{2}) + 2 k (10 k) + 2 k (25) = 0$

Factor $2 k$ out of $2 k (k^{2}) + 2 k (10 k)$ .

$2 k (k^{2} + 10 k) + 2 k (25) = 0$

Factor $2 k$ out of $2 k (k^{2} + 10 k) + 2 k (25)$ .

$2 k (k^{2} + 10 k + 25) = 0$

Factor using the perfect square rule.

Rewrite $25$ as $5^{2}$ .

$2 k (k^{2} + 10 k + 5^{2}) = 0$

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$10 k = 2 \cdot k \cdot 5$

Rewrite the polynomial.

$2 k (k^{2} + 2 \cdot k \cdot 5 + 5^{2}) = 0$

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = k$ and $b = 5$ .

$2 k {(k + 5)}^{2} = 0$

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$k = 0$

${(k + 5)}^{2} = 0$

Set $k$ equal to $0$ .

$k = 0$

Set ${(k + 5)}^{2}$ equal to $0$ and solve for $k$ .

Set ${(k + 5)}^{2}$ equal to $0$ .

${(k + 5)}^{2} = 0$

Solve ${(k + 5)}^{2} = 0$ for $k$ .

Set the $k + 5$ equal to $0$ .

$k + 5 = 0$

Subtract $5$ from both sides of the equation.

$k = - 5$

The final solution is all the values that make $2 k {(k + 5)}^{2} = 0$ true.

$k = 0, - 5$

Step 5

The equation is undefined where the denominator equals $0$ , the argument of a square root is less than $0$ , or the argument of a logarithm is less than or equal to $0$ .

$k = - 5, k = 0$

Step 6

Find Where Undefined/Discontinuous (6k+30)/(33k^5)*(k^2+16k+55)/(2k^3+20k^2+50k)

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