Find Where Undefined/Discontinuous (3y^3-27y)/(y^2-9y+18)*(8y^2)/(3y^2-6y-45)

$\frac{3 y^{3} - 27 y}{y^{2} - 9 y + 18} \cdot \frac{8 y^{2}}{3 y^{2} - 6 y - 45}$

Step 1

Set the denominator in $\frac{3 y^{3} - 27 y}{y^{2} - 9 y + 18}$ equal to $0$ to find where the expression is undefined.

$y^{2} - 9 y + 18 = 0$

Step 2

Solve for $y$ .

Factor $y^{2} - 9 y + 18$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $18$ and whose sum is $- 9$ .

$- 6, - 3$

Write the factored form using these integers.

$(y - 6) (y - 3) = 0$

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

$y - 6 = 0$

$y - 3 = 0$

Set $y - 6$ equal to $0$ and solve for $y$ .

Set $y - 6$ equal to $0$ .

$y - 6 = 0$

Add $6$ to both sides of the equation.

$y = 6$

Set $y - 3$ equal to $0$ and solve for $y$ .

Set $y - 3$ equal to $0$ .

$y - 3 = 0$

Add $3$ to both sides of the equation.

$y = 3$

The final solution is all the values that make $(y - 6) (y - 3) = 0$ true.

$y = 6, 3$

Step 3

Set the denominator in $\frac{8 y^{2}}{3 y^{2} - 6 y - 45}$ equal to $0$ to find where the expression is undefined.

$3 y^{2} - 6 y - 45 = 0$

Step 4

Solve for $y$ .

Factor the left side of the equation.

Factor $3$ out of $3 y^{2} - 6 y - 45$ .

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

$3 (y^{2}) - 6 y - 45 = 0$

Factor $3$ out of $- 6 y$ .

$3 (y^{2}) + 3 (- 2 y) - 45 = 0$

Factor $3$ out of $- 45$ .

$3 y^{2} + 3 (- 2 y) + 3 \cdot - 15 = 0$

Factor $3$ out of $3 y^{2} + 3 (- 2 y)$ .

$3 (y^{2} - 2 y) + 3 \cdot - 15 = 0$

Factor $3$ out of $3 (y^{2} - 2 y) + 3 \cdot - 15$ .

$3 (y^{2} - 2 y - 15) = 0$

Factor.

Factor $y^{2} - 2 y - 15$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 15$ and whose sum is $- 2$ .

$- 5, 3$

Write the factored form using these integers.

$3 ((y - 5) (y + 3)) = 0$

Remove unnecessary parentheses.

$3 (y - 5) (y + 3) = 0$

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

$y - 5 = 0$

$y + 3 = 0$

Set $y - 5$ equal to $0$ and solve for $y$ .

Set $y - 5$ equal to $0$ .

$y - 5 = 0$

Add $5$ to both sides of the equation.

$y = 5$

Set $y + 3$ equal to $0$ and solve for $y$ .

Set $y + 3$ equal to $0$ .

$y + 3 = 0$

Subtract $3$ from both sides of the equation.

$y = - 3$

The final solution is all the values that make $3 (y - 5) (y + 3) = 0$ true.

$y = 5, - 3$

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 $0$ .

$y = - 3, y = 3, y = 5, y = 6$

Step 6

Find Where Undefined/Discontinuous (3y^3-27y)/(y^2-9y+18)*(8y^2)/(3y^2-6y-45)

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