Solve (n^2 - 18 - 3n)/(n^3)

Question

\frac{n ^{2} - 18 - 3 n}{n ^{3}}

Find the excluded values

n = 0

Evaluate

\frac{n ^{2} - 18 - 3 n}{n ^{3}}

To find the excluded values,set the denominators equal to 0

n^{3} = 0

Solution

n = 0

Show Solution

- \frac{18}{n ^{3}} - \frac{3}{n ^{2}} + \frac{1}{n}

Evaluate

\frac{n ^{2} - 18 - 3 n}{n ^{3}}

For each factor in the denominator,write a new fraction

\frac{?}{n ^{3}} + \frac{?}{n ^{2}} + \frac{?}{n}

Write the terms in the numerator

\frac{A}{n ^{3}} + \frac{B}{n ^{2}} + \frac{C}{n}

Set the sum of fractions equal to the original fraction

\frac{n ^{2} - 18 - 3 n}{n ^{3}} = \frac{A}{n ^{3}} + \frac{B}{n ^{2}} + \frac{C}{n}

Multiply both sides

\frac{n ^{2} - 18 - 3 n}{n ^{3}} \times n^{3} = \frac{A}{n ^{3}} \times n^{3} + \frac{B}{n ^{2}} \times n^{3} + \frac{C}{n} \times n^{3}

Simplify the expression

n^{2} - 18 - 3 n = 1 \times A + n B + n^{2} C

Any expression multiplied by 1 remains the same

n^{2} - 18 - 3 n = A + n B + n^{2} C

Group the terms

n^{2} - 18 - 3 n = C n^{2} + B n + A

Equate the coefficients

⎩ ⎨ ⎧ 1 = C - 3 = B - 18 = A

Swap the sides

⎩ ⎨ ⎧ C = 1 B = - 3 A = - 18

Find the intersection

⎩ ⎨ ⎧ A = - 18 B = - 3 C = 1

Solution

- \frac{18}{n ^{3}} - \frac{3}{n ^{2}} + \frac{1}{n}

Show Solution

n_{1} = - 3, n_{2} = 6

Evaluate

\frac{n ^{2} - 18 - 3 n}{n ^{3}}

To find the roots of the expression,set the expression equal to 0

\frac{n ^{2} - 18 - 3 n}{n ^{3}} = 0

The only way a power can not be 0 is when the base not equals 0

\frac{n ^{2} - 18 - 3 n}{n ^{3}} = 0, n \neq = 0

Calculate

\frac{n ^{2} - 18 - 3 n}{n ^{3}} = 0

Cross multiply

n^{2} - 18 - 3 n = n^{3} \times 0

Simplify the equation

n^{2} - 18 - 3 n = 0

Factor the expression

More Steps

Evaluate

n^{2} - 18 - 3 n

Reorder the terms

n^{2} - 3 n - 18

Rewrite the expression

n^{2} + (3 - 6) n - 18

Calculate

n^{2} + 3 n - 6 n - 18

Rewrite the expression

n \times n + n \times 3 - 6 n - 6 \times 3

Factor out n from the expression

n (n + 3) - 6 n - 6 \times 3

Factor out - 6 from the expression

n (n + 3) - 6 (n + 3)

Factor out n + 3 from the expression

(n - 6) (n + 3)

(n - 6) (n + 3) = 0

When the product of factors equals 0,at least one factor is 0

n - 6 = 0 n + 3 = 0

Solve the equation for n

More Steps

Evaluate

n - 6 = 0

Move the constant to the right-hand side and change its sign

n = 0 + 6

Removing 0 doesn't change the value,so remove it from the expression

n = 6

n = 6 n + 3 = 0

Solve the equation for n

More Steps

Evaluate

n + 3 = 0

Move the constant to the right-hand side and change its sign

n = 0 - 3

Removing 0 doesn't change the value,so remove it from the expression

n = - 3

n = 6 n = - 3

Check if the solution is in the defined range

n = 6 n = - 3, n \neq = 0

Find the intersection of the solution and the defined range

n = 6 n = - 3

Solution

n_{1} = - 3, n_{2} = 6

Show Solution