Precalculus Examples

$y = - \frac{6}{20} \cdot ((x + 5) (x + 1) {(x - 2)}^{2})$

Step 1

Find the x-intercepts.

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Step 1.1

To find the x-intercept(s), substitute in $0$ for $y$ and solve for $x$ .

$0 = - \frac{6}{20} \cdot ((x + 5) (x + 1) {(x - 2)}^{2})$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $- \frac{6}{20} \cdot ((x + 5) (x + 1) {(x - 2)}^{2}) = 0$ .

$- \frac{6}{20} \cdot ((x + 5) (x + 1) {(x - 2)}^{2}) = 0$

Step 1.2.2

Multiply both sides of the equation by $- \frac{20}{6}$ .

$- \frac{20}{6} (- \frac{6}{20} \cdot ((x + 5) (x + 1) {(x - 2)}^{2})) = - \frac{20}{6} \cdot 0$

Step 1.2.3

Simplify both sides of the equation.

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Step 1.2.3.1

Simplify the left side.

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Step 1.2.3.1.1

Simplify $- \frac{20}{6} (- \frac{6}{20} \cdot ((x + 5) (x + 1) {(x - 2)}^{2}))$ .

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Step 1.2.3.1.1.1

Reduce the expression by cancelling the common factors.

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Step 1.2.3.1.1.1.1

Cancel the common factor of $20$ and $6$ .

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Step 1.2.3.1.1.1.1.1

Factor $2$ out of $20$ .

$- \frac{2 (10)}{6} (- \frac{6}{20} \cdot ((x + 5) (x + 1) {(x - 2)}^{2})) = - \frac{20}{6} \cdot 0$

Step 1.2.3.1.1.1.1.2

Cancel the common factors.

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Step 1.2.3.1.1.1.1.2.1

Factor $2$ out of $6$ .

$- \frac{2 \cdot 10}{2 \cdot 3} (- \frac{6}{20} \cdot ((x + 5) (x + 1) {(x - 2)}^{2})) = - \frac{20}{6} \cdot 0$

Step 1.2.3.1.1.1.1.2.2

Cancel the common factor.

$- \frac{2 \cdot 10}{2 \cdot 3} (- \frac{6}{20} \cdot ((x + 5) (x + 1) {(x - 2)}^{2})) = - \frac{20}{6} \cdot 0$

Step 1.2.3.1.1.1.1.2.3

Rewrite the expression.

$- \frac{10}{3} (- \frac{6}{20} \cdot ((x + 5) (x + 1) {(x - 2)}^{2})) = - \frac{20}{6} \cdot 0$

Step 1.2.3.1.1.1.2

Cancel the common factor of $6$ and $20$ .

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Step 1.2.3.1.1.1.2.1

Factor $2$ out of $6$ .

$- \frac{10}{3} (- \frac{2 (3)}{20} \cdot ((x + 5) (x + 1) {(x - 2)}^{2})) = - \frac{20}{6} \cdot 0$

Step 1.2.3.1.1.1.2.2

Cancel the common factors.

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Step 1.2.3.1.1.1.2.2.1

Factor $2$ out of $20$ .

$- \frac{10}{3} (- \frac{2 \cdot 3}{2 \cdot 10} \cdot ((x + 5) (x + 1) {(x - 2)}^{2})) = - \frac{20}{6} \cdot 0$

Step 1.2.3.1.1.1.2.2.2

Cancel the common factor.

$- \frac{10}{3} (- \frac{2 \cdot 3}{2 \cdot 10} \cdot ((x + 5) (x + 1) {(x - 2)}^{2})) = - \frac{20}{6} \cdot 0$

Step 1.2.3.1.1.1.2.2.3

Rewrite the expression.

$- \frac{10}{3} (- \frac{3}{10} \cdot ((x + 5) (x + 1) {(x - 2)}^{2})) = - \frac{20}{6} \cdot 0$

Step 1.2.3.1.1.2

Expand $(x + 5) (x + 1)$ using the FOIL Method.

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Step 1.2.3.1.1.2.1

Apply the distributive property.

$- \frac{10}{3} (- \frac{3}{10} \cdot ((x (x + 1) + 5 (x + 1)) {(x - 2)}^{2})) = - \frac{20}{6} \cdot 0$

Step 1.2.3.1.1.2.2

Apply the distributive property.

$- \frac{10}{3} (- \frac{3}{10} \cdot ((x \cdot x + x \cdot 1 + 5 (x + 1)) {(x - 2)}^{2})) = - \frac{20}{6} \cdot 0$

Step 1.2.3.1.1.2.3

Apply the distributive property.

$- \frac{10}{3} (- \frac{3}{10} \cdot ((x \cdot x + x \cdot 1 + 5 x + 5 \cdot 1) {(x - 2)}^{2})) = - \frac{20}{6} \cdot 0$

Step 1.2.3.1.1.3

Simplify and combine like terms.

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Step 1.2.3.1.1.3.1

Simplify each term.

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Step 1.2.3.1.1.3.1.1

Multiply $x$ by $x$ .

$- \frac{10}{3} (- \frac{3}{10} \cdot ((x^{2} + x \cdot 1 + 5 x + 5 \cdot 1) {(x - 2)}^{2})) = - \frac{20}{6} \cdot 0$

Step 1.2.3.1.1.3.1.2

Multiply $x$ by $1$ .

$- \frac{10}{3} (- \frac{3}{10} \cdot ((x^{2} + x + 5 x + 5 \cdot 1) {(x - 2)}^{2})) = - \frac{20}{6} \cdot 0$

Step 1.2.3.1.1.3.1.3

Multiply $5$ by $1$ .

$- \frac{10}{3} (- \frac{3}{10} \cdot ((x^{2} + x + 5 x + 5) {(x - 2)}^{2})) = - \frac{20}{6} \cdot 0$

Step 1.2.3.1.1.3.2

Add $x$ and $5 x$ .

$- \frac{10}{3} (- \frac{3}{10} \cdot ((x^{2} + 6 x + 5) {(x - 2)}^{2})) = - \frac{20}{6} \cdot 0$

Step 1.2.3.1.1.4

Apply the distributive property.

$- \frac{10}{3} (- \frac{3}{10} \cdot (x^{2} {(x - 2)}^{2} + 6 x {(x - 2)}^{2} + 5 {(x - 2)}^{2})) = - \frac{20}{6} \cdot 0$

Step 1.2.3.1.1.5

Apply the distributive property.

$- \frac{10}{3} (- \frac{3}{10} (x^{2} {(x - 2)}^{2}) - \frac{3}{10} (6 x {(x - 2)}^{2}) - \frac{3}{10} (5 {(x - 2)}^{2})) = - \frac{20}{6} \cdot 0$

Step 1.2.3.1.1.6

Simplify.

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Step 1.2.3.1.1.6.1

Multiply $- \frac{3}{10} (x^{2} {(x - 2)}^{2})$ .

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Step 1.2.3.1.1.6.1.1

Combine $x^{2}$ and $\frac{3}{10}$ .

$- \frac{10}{3} (- \frac{x^{2} \cdot 3}{10} {(x - 2)}^{2} - \frac{3}{10} (6 x {(x - 2)}^{2}) - \frac{3}{10} (5 {(x - 2)}^{2})) = - \frac{20}{6} \cdot 0$

Step 1.2.3.1.1.6.1.2

Combine ${(x - 2)}^{2}$ and $\frac{x^{2} \cdot 3}{10}$ .

$- \frac{10}{3} (- \frac{{(x - 2)}^{2} (x^{2} \cdot 3)}{10} - \frac{3}{10} (6 x {(x - 2)}^{2}) - \frac{3}{10} (5 {(x - 2)}^{2})) = - \frac{20}{6} \cdot 0$

Step 1.2.3.1.1.6.2

Cancel the common factor of $2$ .

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Step 1.2.3.1.1.6.2.1

Move the leading negative in $- \frac{3}{10}$ into the numerator.

$- \frac{10}{3} (- \frac{{(x - 2)}^{2} (x^{2} \cdot 3)}{10} + \frac{- 3}{10} (6 x {(x - 2)}^{2}) - \frac{3}{10} (5 {(x - 2)}^{2})) = - \frac{20}{6} \cdot 0$

Step 1.2.3.1.1.6.2.2

Factor $2$ out of $10$ .

$- \frac{10}{3} (- \frac{{(x - 2)}^{2} (x^{2} \cdot 3)}{10} + \frac{- 3}{2 (5)} (6 x {(x - 2)}^{2}) - \frac{3}{10} (5 {(x - 2)}^{2})) = - \frac{20}{6} \cdot 0$

Step 1.2.3.1.1.6.2.3

Factor $2$ out of $6 x {(x - 2)}^{2}$ .

$- \frac{10}{3} (- \frac{{(x - 2)}^{2} (x^{2} \cdot 3)}{10} + \frac{- 3}{2 (5)} (2 (3 x {(x - 2)}^{2})) - \frac{3}{10} (5 {(x - 2)}^{2})) = - \frac{20}{6} \cdot 0$

Step 1.2.3.1.1.6.2.4

Cancel the common factor.

$- \frac{10}{3} (- \frac{{(x - 2)}^{2} (x^{2} \cdot 3)}{10} + \frac{- 3}{2 \cdot 5} (2 (3 x {(x - 2)}^{2})) - \frac{3}{10} (5 {(x - 2)}^{2})) = - \frac{20}{6} \cdot 0$

Step 1.2.3.1.1.6.2.5

Rewrite the expression.

$- \frac{10}{3} (- \frac{{(x - 2)}^{2} (x^{2} \cdot 3)}{10} + \frac{- 3}{5} (3 x {(x - 2)}^{2}) - \frac{3}{10} (5 {(x - 2)}^{2})) = - \frac{20}{6} \cdot 0$

Step 1.2.3.1.1.6.3

Combine $3$ and $\frac{- 3}{5}$ .

$- \frac{10}{3} (- \frac{{(x - 2)}^{2} (x^{2} \cdot 3)}{10} + \frac{3 \cdot - 3}{5} (x {(x - 2)}^{2}) - \frac{3}{10} (5 {(x - 2)}^{2})) = - \frac{20}{6} \cdot 0$

Step 1.2.3.1.1.6.4

Multiply $3$ by $- 3$ .

$- \frac{10}{3} (- \frac{{(x - 2)}^{2} (x^{2} \cdot 3)}{10} + \frac{- 9}{5} (x {(x - 2)}^{2}) - \frac{3}{10} (5 {(x - 2)}^{2})) = - \frac{20}{6} \cdot 0$

Step 1.2.3.1.1.6.5

Combine $x$ and $\frac{- 9}{5}$ .

$- \frac{10}{3} (- \frac{{(x - 2)}^{2} (x^{2} \cdot 3)}{10} + \frac{x \cdot - 9}{5} {(x - 2)}^{2} - \frac{3}{10} (5 {(x - 2)}^{2})) = - \frac{20}{6} \cdot 0$

Step 1.2.3.1.1.6.6

Combine $\frac{x \cdot - 9}{5}$ and ${(x - 2)}^{2}$ .

$- \frac{10}{3} (- \frac{{(x - 2)}^{2} (x^{2} \cdot 3)}{10} + \frac{x \cdot - 9 {(x - 2)}^{2}}{5} - \frac{3}{10} (5 {(x - 2)}^{2})) = - \frac{20}{6} \cdot 0$

Step 1.2.3.1.1.6.7

Cancel the common factor of $5$ .

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Step 1.2.3.1.1.6.7.1

Move the leading negative in $- \frac{3}{10}$ into the numerator.

$- \frac{10}{3} (- \frac{{(x - 2)}^{2} (x^{2} \cdot 3)}{10} + \frac{x \cdot - 9 {(x - 2)}^{2}}{5} + \frac{- 3}{10} (5 {(x - 2)}^{2})) = - \frac{20}{6} \cdot 0$

Step 1.2.3.1.1.6.7.2

Factor $5$ out of $10$ .

$- \frac{10}{3} (- \frac{{(x - 2)}^{2} (x^{2} \cdot 3)}{10} + \frac{x \cdot - 9 {(x - 2)}^{2}}{5} + \frac{- 3}{5 (2)} (5 {(x - 2)}^{2})) = - \frac{20}{6} \cdot 0$

Step 1.2.3.1.1.6.7.3

Factor $5$ out of $5 {(x - 2)}^{2}$ .

$- \frac{10}{3} (- \frac{{(x - 2)}^{2} (x^{2} \cdot 3)}{10} + \frac{x \cdot - 9 {(x - 2)}^{2}}{5} + \frac{- 3}{5 (2)} (5 ({(x - 2)}^{2}))) = - \frac{20}{6} \cdot 0$

Step 1.2.3.1.1.6.7.4

Cancel the common factor.

$- \frac{10}{3} (- \frac{{(x - 2)}^{2} (x^{2} \cdot 3)}{10} + \frac{x \cdot - 9 {(x - 2)}^{2}}{5} + \frac{- 3}{5 \cdot 2} (5 {(x - 2)}^{2})) = - \frac{20}{6} \cdot 0$

Step 1.2.3.1.1.6.7.5

Rewrite the expression.

$- \frac{10}{3} (- \frac{{(x - 2)}^{2} (x^{2} \cdot 3)}{10} + \frac{x \cdot - 9 {(x - 2)}^{2}}{5} + \frac{- 3}{2} {(x - 2)}^{2}) = - \frac{20}{6} \cdot 0$

Step 1.2.3.1.1.6.8

Combine $\frac{- 3}{2}$ and ${(x - 2)}^{2}$ .

$- \frac{10}{3} (- \frac{{(x - 2)}^{2} (x^{2} \cdot 3)}{10} + \frac{x \cdot - 9 {(x - 2)}^{2}}{5} + \frac{- 3 {(x - 2)}^{2}}{2}) = - \frac{20}{6} \cdot 0$

Step 1.2.3.1.1.7

Simplify each term.

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Step 1.2.3.1.1.7.1

Move $3$ to the left of ${(x - 2)}^{2} x^{2}$ .

$- \frac{10}{3} (- \frac{3 \cdot ({(x - 2)}^{2} x^{2})}{10} + \frac{x \cdot - 9 {(x - 2)}^{2}}{5} + \frac{- 3 {(x - 2)}^{2}}{2}) = - \frac{20}{6} \cdot 0$

Step 1.2.3.1.1.7.2

Move $- 9$ to the left of $x$ .

$- \frac{10}{3} (- \frac{3 {(x - 2)}^{2} x^{2}}{10} + \frac{- 9 \cdot x {(x - 2)}^{2}}{5} + \frac{- 3 {(x - 2)}^{2}}{2}) = - \frac{20}{6} \cdot 0$

Step 1.2.3.1.1.7.3

Move the negative in front of the fraction.

$- \frac{10}{3} (- \frac{3 {(x - 2)}^{2} x^{2}}{10} - \frac{9 \cdot x {(x - 2)}^{2}}{5} + \frac{- 3 {(x - 2)}^{2}}{2}) = - \frac{20}{6} \cdot 0$

Step 1.2.3.1.1.7.4

Move the negative in front of the fraction.

$- \frac{10}{3} (- \frac{3 {(x - 2)}^{2} x^{2}}{10} - \frac{9 x {(x - 2)}^{2}}{5} - \frac{3 {(x - 2)}^{2}}{2}) = - \frac{20}{6} \cdot 0$

Step 1.2.3.1.1.8

To write $- \frac{9 x {(x - 2)}^{2}}{5}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- \frac{10}{3} (- \frac{3 {(x - 2)}^{2} x^{2}}{10} - \frac{9 x {(x - 2)}^{2}}{5} \cdot \frac{2}{2} - \frac{3 {(x - 2)}^{2}}{2}) = - \frac{20}{6} \cdot 0$

Step 1.2.3.1.1.9

Write each expression with a common denominator of $10$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.2.3.1.1.9.1

Multiply $\frac{9 x {(x - 2)}^{2}}{5}$ by $\frac{2}{2}$ .

$- \frac{10}{3} (- \frac{3 {(x - 2)}^{2} x^{2}}{10} - \frac{9 x {(x - 2)}^{2} \cdot 2}{5 \cdot 2} - \frac{3 {(x - 2)}^{2}}{2}) = - \frac{20}{6} \cdot 0$

Step 1.2.3.1.1.9.2

Multiply $5$ by $2$ .

$- \frac{10}{3} (- \frac{3 {(x - 2)}^{2} x^{2}}{10} - \frac{9 x {(x - 2)}^{2} \cdot 2}{10} - \frac{3 {(x - 2)}^{2}}{2}) = - \frac{20}{6} \cdot 0$

Step 1.2.3.1.1.10

Combine the numerators over the common denominator.

$- \frac{10}{3} (\frac{- 3 {(x - 2)}^{2} x^{2} - 9 x {(x - 2)}^{2} \cdot 2}{10} - \frac{3 {(x - 2)}^{2}}{2}) = - \frac{20}{6} \cdot 0$

Step 1.2.3.1.1.11

Simplify the numerator.

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Step 1.2.3.1.1.11.1

Factor $3 {(x - 2)}^{2} x$ out of $- 3 {(x - 2)}^{2} x^{2} - 9 x {(x - 2)}^{2} \cdot 2$ .

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Step 1.2.3.1.1.11.1.1

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

$- \frac{10}{3} (\frac{3 {(x - 2)}^{2} x (- x) - 9 x {(x - 2)}^{2} \cdot 2}{10} - \frac{3 {(x - 2)}^{2}}{2}) = - \frac{20}{6} \cdot 0$

Step 1.2.3.1.1.11.1.2

Factor $3 {(x - 2)}^{2} x$ out of $- 9 x {(x - 2)}^{2} \cdot 2$ .

$- \frac{10}{3} (\frac{3 {(x - 2)}^{2} x (- x) + 3 {(x - 2)}^{2} x (- 3 \cdot 2)}{10} - \frac{3 {(x - 2)}^{2}}{2}) = - \frac{20}{6} \cdot 0$

Step 1.2.3.1.1.11.1.3

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

$- \frac{10}{3} (\frac{3 {(x - 2)}^{2} x (- x - 3 \cdot 2)}{10} - \frac{3 {(x - 2)}^{2}}{2}) = - \frac{20}{6} \cdot 0$

Step 1.2.3.1.1.11.2

Multiply $- 3$ by $2$ .

$- \frac{10}{3} (\frac{3 {(x - 2)}^{2} x (- x - 6)}{10} - \frac{3 {(x - 2)}^{2}}{2}) = - \frac{20}{6} \cdot 0$

Step 1.2.3.1.1.12

To write $- \frac{3 {(x - 2)}^{2}}{2}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$- \frac{10}{3} (\frac{3 {(x - 2)}^{2} x (- x - 6)}{10} - \frac{3 {(x - 2)}^{2}}{2} \cdot \frac{5}{5}) = - \frac{20}{6} \cdot 0$

Step 1.2.3.1.1.13

Write each expression with a common denominator of $10$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.2.3.1.1.13.1

Multiply $\frac{3 {(x - 2)}^{2}}{2}$ by $\frac{5}{5}$ .

$- \frac{10}{3} (\frac{3 {(x - 2)}^{2} x (- x - 6)}{10} - \frac{3 {(x - 2)}^{2} \cdot 5}{2 \cdot 5}) = - \frac{20}{6} \cdot 0$

Step 1.2.3.1.1.13.2

Multiply $2$ by $5$ .

$- \frac{10}{3} (\frac{3 {(x - 2)}^{2} x (- x - 6)}{10} - \frac{3 {(x - 2)}^{2} \cdot 5}{10}) = - \frac{20}{6} \cdot 0$

Step 1.2.3.1.1.14

Simplify terms.

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Step 1.2.3.1.1.14.1

Combine the numerators over the common denominator.

$- \frac{10}{3} \cdot \frac{3 {(x - 2)}^{2} x (- x - 6) - 3 {(x - 2)}^{2} \cdot 5}{10} = - \frac{20}{6} \cdot 0$

Step 1.2.3.1.1.14.2

Cancel the common factor of $10$ .

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Step 1.2.3.1.1.14.2.1

Move the leading negative in $- \frac{10}{3}$ into the numerator.

$\frac{- 10}{3} \cdot \frac{3 {(x - 2)}^{2} x (- x - 6) - 3 {(x - 2)}^{2} \cdot 5}{10} = - \frac{20}{6} \cdot 0$

Step 1.2.3.1.1.14.2.2

Factor $10$ out of $- 10$ .

$\frac{10 (- 1)}{3} \cdot \frac{3 {(x - 2)}^{2} x (- x - 6) - 3 {(x - 2)}^{2} \cdot 5}{10} = - \frac{20}{6} \cdot 0$

Step 1.2.3.1.1.14.2.3

Cancel the common factor.

$\frac{10 \cdot - 1}{3} \cdot \frac{3 {(x - 2)}^{2} x (- x - 6) - 3 {(x - 2)}^{2} \cdot 5}{10} = - \frac{20}{6} \cdot 0$

Step 1.2.3.1.1.14.2.4

Rewrite the expression.

$\frac{- 1}{3} (3 {(x - 2)}^{2} x (- x - 6) - 3 {(x - 2)}^{2} \cdot 5) = - \frac{20}{6} \cdot 0$

Step 1.2.3.1.1.14.3

Simplify the expression.

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Step 1.2.3.1.1.14.3.1

Multiply $5$ by $- 3$ .

$\frac{- 1}{3} (3 {(x - 2)}^{2} x (- x - 6) - 15 {(x - 2)}^{2}) = - \frac{20}{6} \cdot 0$

Step 1.2.3.1.1.14.3.2

Move the negative in front of the fraction.

$- \frac{1}{3} (3 {(x - 2)}^{2} x (- x - 6) - 15 {(x - 2)}^{2}) = - \frac{20}{6} \cdot 0$

Step 1.2.3.1.1.15

Simplify each term.

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Step 1.2.3.1.1.15.1

Apply the distributive property.

$- \frac{1}{3} (3 {(x - 2)}^{2} x (- x) + 3 {(x - 2)}^{2} x \cdot - 6 - 15 {(x - 2)}^{2}) = - \frac{20}{6} \cdot 0$

Step 1.2.3.1.1.15.2

Multiply $x$ by $x$ by adding the exponents.

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Step 1.2.3.1.1.15.2.1

Move $x$ .

$- \frac{1}{3} (3 {(x - 2)}^{2} (x \cdot x) \cdot - 1 + 3 {(x - 2)}^{2} x \cdot - 6 - 15 {(x - 2)}^{2}) = - \frac{20}{6} \cdot 0$

Step 1.2.3.1.1.15.2.2

Multiply $x$ by $x$ .

$- \frac{1}{3} (3 {(x - 2)}^{2} x^{2} \cdot - 1 + 3 {(x - 2)}^{2} x \cdot - 6 - 15 {(x - 2)}^{2}) = - \frac{20}{6} \cdot 0$

Step 1.2.3.1.1.15.3

Multiply $- 6$ by $3$ .

$- \frac{1}{3} (3 {(x - 2)}^{2} x^{2} \cdot - 1 - 18 {(x - 2)}^{2} x - 15 {(x - 2)}^{2}) = - \frac{20}{6} \cdot 0$

Step 1.2.3.1.1.15.4

Multiply $- 1$ by $3$ .

$- \frac{1}{3} (- 3 {(x - 2)}^{2} x^{2} - 18 {(x - 2)}^{2} x - 15 {(x - 2)}^{2}) = - \frac{20}{6} \cdot 0$

Step 1.2.3.1.1.16

Apply the distributive property.

$- \frac{1}{3} (- 3 {(x - 2)}^{2} x^{2}) - \frac{1}{3} (- 18 {(x - 2)}^{2} x) - \frac{1}{3} (- 15 {(x - 2)}^{2}) = - \frac{20}{6} \cdot 0$

Step 1.2.3.1.1.17

Simplify.

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Step 1.2.3.1.1.17.1

Cancel the common factor of $3$ .

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Step 1.2.3.1.1.17.1.1

Move the leading negative in $- \frac{1}{3}$ into the numerator.

$\frac{- 1}{3} (- 3 {(x - 2)}^{2} x^{2}) - \frac{1}{3} (- 18 {(x - 2)}^{2} x) - \frac{1}{3} (- 15 {(x - 2)}^{2}) = - \frac{20}{6} \cdot 0$

Step 1.2.3.1.1.17.1.2

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

$\frac{- 1}{3} (3 (- {(x - 2)}^{2} x^{2})) - \frac{1}{3} (- 18 {(x - 2)}^{2} x) - \frac{1}{3} (- 15 {(x - 2)}^{2}) = - \frac{20}{6} \cdot 0$

Step 1.2.3.1.1.17.1.3

Cancel the common factor.

$\frac{- 1}{3} (3 (- {(x - 2)}^{2} x^{2})) - \frac{1}{3} (- 18 {(x - 2)}^{2} x) - \frac{1}{3} (- 15 {(x - 2)}^{2}) = - \frac{20}{6} \cdot 0$

Step 1.2.3.1.1.17.1.4

Rewrite the expression.

$- (- {(x - 2)}^{2} x^{2}) - \frac{1}{3} (- 18 {(x - 2)}^{2} x) - \frac{1}{3} (- 15 {(x - 2)}^{2}) = - \frac{20}{6} \cdot 0$

Step 1.2.3.1.1.17.2

Multiply $- 1$ by $- 1$ .

$1 ({(x - 2)}^{2} x^{2}) - \frac{1}{3} (- 18 {(x - 2)}^{2} x) - \frac{1}{3} (- 15 {(x - 2)}^{2}) = - \frac{20}{6} \cdot 0$

Step 1.2.3.1.1.17.3

Multiply ${(x - 2)}^{2}$ by $1$ .

${(x - 2)}^{2} x^{2} - \frac{1}{3} (- 18 {(x - 2)}^{2} x) - \frac{1}{3} (- 15 {(x - 2)}^{2}) = - \frac{20}{6} \cdot 0$

Step 1.2.3.1.1.17.4

Cancel the common factor of $3$ .

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Step 1.2.3.1.1.17.4.1

Move the leading negative in $- \frac{1}{3}$ into the numerator.

${(x - 2)}^{2} x^{2} + \frac{- 1}{3} (- 18 {(x - 2)}^{2} x) - \frac{1}{3} (- 15 {(x - 2)}^{2}) = - \frac{20}{6} \cdot 0$

Step 1.2.3.1.1.17.4.2

Factor $3$ out of $- 18 {(x - 2)}^{2} x$ .

${(x - 2)}^{2} x^{2} + \frac{- 1}{3} (3 (- 6 {(x - 2)}^{2} x)) - \frac{1}{3} (- 15 {(x - 2)}^{2}) = - \frac{20}{6} \cdot 0$

Step 1.2.3.1.1.17.4.3

Cancel the common factor.

${(x - 2)}^{2} x^{2} + \frac{- 1}{3} (3 (- 6 {(x - 2)}^{2} x)) - \frac{1}{3} (- 15 {(x - 2)}^{2}) = - \frac{20}{6} \cdot 0$

Step 1.2.3.1.1.17.4.4

Rewrite the expression.

${(x - 2)}^{2} x^{2} - (- 6 {(x - 2)}^{2} x) - \frac{1}{3} (- 15 {(x - 2)}^{2}) = - \frac{20}{6} \cdot 0$

Step 1.2.3.1.1.17.5

Multiply $- 6$ by $- 1$ .

${(x - 2)}^{2} x^{2} + 6 ({(x - 2)}^{2} x) - \frac{1}{3} (- 15 {(x - 2)}^{2}) = - \frac{20}{6} \cdot 0$

Step 1.2.3.1.1.17.6

Cancel the common factor of $3$ .

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Step 1.2.3.1.1.17.6.1

Move the leading negative in $- \frac{1}{3}$ into the numerator.

${(x - 2)}^{2} x^{2} + 6 ({(x - 2)}^{2} x) + \frac{- 1}{3} (- 15 {(x - 2)}^{2}) = - \frac{20}{6} \cdot 0$

Step 1.2.3.1.1.17.6.2

Factor $3$ out of $- 15 {(x - 2)}^{2}$ .

${(x - 2)}^{2} x^{2} + 6 ({(x - 2)}^{2} x) + \frac{- 1}{3} (3 (- 5 {(x - 2)}^{2})) = - \frac{20}{6} \cdot 0$

Step 1.2.3.1.1.17.6.3

Cancel the common factor.

${(x - 2)}^{2} x^{2} + 6 ({(x - 2)}^{2} x) + \frac{- 1}{3} (3 (- 5 {(x - 2)}^{2})) = - \frac{20}{6} \cdot 0$

Step 1.2.3.1.1.17.6.4

Rewrite the expression.

${(x - 2)}^{2} x^{2} + 6 ({(x - 2)}^{2} x) - (- 5 {(x - 2)}^{2}) = - \frac{20}{6} \cdot 0$

Step 1.2.3.1.1.17.7

Multiply $- 5$ by $- 1$ .

${(x - 2)}^{2} x^{2} + 6 ({(x - 2)}^{2} x) + 5 {(x - 2)}^{2} = - \frac{20}{6} \cdot 0$

Step 1.2.3.1.1.18

Reorder factors in ${(x - 2)}^{2} x^{2} + 6 {(x - 2)}^{2} x + 5 {(x - 2)}^{2}$ .

$x^{2} {(x - 2)}^{2} + 6 x {(x - 2)}^{2} + 5 {(x - 2)}^{2} = - \frac{20}{6} \cdot 0$

Step 1.2.3.2

Simplify the right side.

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Step 1.2.3.2.1

Simplify $- \frac{20}{6} \cdot 0$ .

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Step 1.2.3.2.1.1

Cancel the common factor of $20$ and $6$ .

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Step 1.2.3.2.1.1.1

Factor $2$ out of $20$ .

$x^{2} {(x - 2)}^{2} + 6 x {(x - 2)}^{2} + 5 {(x - 2)}^{2} = - \frac{2 (10)}{6} \cdot 0$

Step 1.2.3.2.1.1.2

Cancel the common factors.

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Step 1.2.3.2.1.1.2.1

Factor $2$ out of $6$ .

$x^{2} {(x - 2)}^{2} + 6 x {(x - 2)}^{2} + 5 {(x - 2)}^{2} = - \frac{2 \cdot 10}{2 \cdot 3} \cdot 0$

Step 1.2.3.2.1.1.2.2

Cancel the common factor.

$x^{2} {(x - 2)}^{2} + 6 x {(x - 2)}^{2} + 5 {(x - 2)}^{2} = - \frac{2 \cdot 10}{2 \cdot 3} \cdot 0$

Step 1.2.3.2.1.1.2.3

Rewrite the expression.

$x^{2} {(x - 2)}^{2} + 6 x {(x - 2)}^{2} + 5 {(x - 2)}^{2} = - \frac{10}{3} \cdot 0$

Step 1.2.3.2.1.2

Multiply $- \frac{10}{3} \cdot 0$ .

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Step 1.2.3.2.1.2.1

Multiply $0$ by $- 1$ .

$x^{2} {(x - 2)}^{2} + 6 x {(x - 2)}^{2} + 5 {(x - 2)}^{2} = 0 (\frac{10}{3})$

Step 1.2.3.2.1.2.2

Multiply $0$ by $\frac{10}{3}$ .

$x^{2} {(x - 2)}^{2} + 6 x {(x - 2)}^{2} + 5 {(x - 2)}^{2} = 0$

Step 1.2.4

Factor the left side of the equation.

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Step 1.2.4.1

Factor ${(x - 2)}^{2}$ out of $x^{2} {(x - 2)}^{2} + 6 x {(x - 2)}^{2} + 5 {(x - 2)}^{2}$ .

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Step 1.2.4.1.1

Factor ${(x - 2)}^{2}$ out of $x^{2} {(x - 2)}^{2}$ .

${(x - 2)}^{2} x^{2} + 6 x {(x - 2)}^{2} + 5 {(x - 2)}^{2} = 0$

Step 1.2.4.1.2

Factor ${(x - 2)}^{2}$ out of $6 x {(x - 2)}^{2}$ .

${(x - 2)}^{2} x^{2} + {(x - 2)}^{2} (6 x) + 5 {(x - 2)}^{2} = 0$

Step 1.2.4.1.3

Factor ${(x - 2)}^{2}$ out of $5 {(x - 2)}^{2}$ .

${(x - 2)}^{2} x^{2} + {(x - 2)}^{2} (6 x) + {(x - 2)}^{2} \cdot 5 = 0$

Step 1.2.4.1.4

Factor ${(x - 2)}^{2}$ out of ${(x - 2)}^{2} x^{2} + {(x - 2)}^{2} (6 x)$ .

${(x - 2)}^{2} (x^{2} + 6 x) + {(x - 2)}^{2} \cdot 5 = 0$

Step 1.2.4.1.5

Factor ${(x - 2)}^{2}$ out of ${(x - 2)}^{2} (x^{2} + 6 x) + {(x - 2)}^{2} \cdot 5$ .

${(x - 2)}^{2} (x^{2} + 6 x + 5) = 0$

Step 1.2.4.2

Factor.

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Step 1.2.4.2.1

Factor $x^{2} + 6 x + 5$ using the AC method.

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Step 1.2.4.2.1.1

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 $5$ and whose sum is $6$ .

$1, 5$

Step 1.2.4.2.1.2

Write the factored form using these integers.

${(x - 2)}^{2} ((x + 1) (x + 5)) = 0$

Step 1.2.4.2.2

Remove unnecessary parentheses.

${(x - 2)}^{2} (x + 1) (x + 5) = 0$

Step 1.2.5

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

${(x - 2)}^{2} = 0$

$x + 1 = 0$

$x + 5 = 0$

Step 1.2.6

Set ${(x - 2)}^{2}$ equal to $0$ and solve for $x$ .

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Step 1.2.6.1

Set ${(x - 2)}^{2}$ equal to $0$ .

${(x - 2)}^{2} = 0$

Step 1.2.6.2

Solve ${(x - 2)}^{2} = 0$ for $x$ .

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Step 1.2.6.2.1

Set the $x - 2$ equal to $0$ .

$x - 2 = 0$

Step 1.2.6.2.2

Add $2$ to both sides of the equation.

$x = 2$

Step 1.2.7

Set $x + 1$ equal to $0$ and solve for $x$ .

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Step 1.2.7.1

Set $x + 1$ equal to $0$ .

$x + 1 = 0$

Step 1.2.7.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 1.2.8

Set $x + 5$ equal to $0$ and solve for $x$ .

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Step 1.2.8.1

Set $x + 5$ equal to $0$ .

$x + 5 = 0$

Step 1.2.8.2

Subtract $5$ from both sides of the equation.

$x = - 5$

Step 1.2.9

The final solution is all the values that make ${(x - 2)}^{2} (x + 1) (x + 5) = 0$ true.

$x = 2, - 1, - 5$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(2, 0), (- 1, 0), (- 5, 0)$

Step 2

Find the y-intercepts.

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Step 2.1

To find the y-intercept(s), substitute in $0$ for $x$ and solve for $y$ .

$y = - \frac{6}{20} \cdot (((0) + 5) ((0) + 1) {((0) - 2)}^{2})$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = - \frac{6}{20} \cdot ((0 + 5) ((0) + 1) {((0) - 2)}^{2})$

Step 2.2.2

Remove parentheses.

$y = - \frac{6}{20} \cdot ((0 + 5) (0 + 1) {((0) - 2)}^{2})$

Step 2.2.3

Remove parentheses.

$y = - \frac{6}{20} \cdot ((0 + 5) (0 + 1) {(0 - 2)}^{2})$

Step 2.2.4

Remove parentheses.

$y = - \frac{6}{20} \cdot (((0 + 5) (0 + 1)) {(0 - 2)}^{2})$

Step 2.2.5

Remove parentheses.

$y = - \frac{6}{20} \cdot ((0 + 5) (0 + 1) {(0 - 2)}^{2})$

Step 2.2.6

Remove parentheses.

$y = - \frac{6}{20} \cdot (((0) + 5) ((0) + 1) {((0) - 2)}^{2})$

Step 2.2.7

Simplify $- \frac{6}{20} \cdot (((0) + 5) ((0) + 1) {((0) - 2)}^{2})$ .

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Step 2.2.7.1

Simplify terms.

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Step 2.2.7.1.1

Cancel the common factor of $6$ and $20$ .

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Step 2.2.7.1.1.1

Factor $2$ out of $6$ .

$y = - \frac{2 (3)}{20} \cdot (((0) + 5) ((0) + 1) {((0) - 2)}^{2})$

Step 2.2.7.1.1.2

Cancel the common factors.

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Step 2.2.7.1.1.2.1

Factor $2$ out of $20$ .

$y = - \frac{2 \cdot 3}{2 \cdot 10} \cdot (((0) + 5) ((0) + 1) {((0) - 2)}^{2})$

Step 2.2.7.1.1.2.2

Cancel the common factor.

$y = - \frac{2 \cdot 3}{2 \cdot 10} \cdot (((0) + 5) ((0) + 1) {((0) - 2)}^{2})$

Step 2.2.7.1.1.2.3

Rewrite the expression.

$y = - \frac{3}{10} \cdot (((0) + 5) ((0) + 1) {((0) - 2)}^{2})$

Step 2.2.7.1.2

Add $0$ and $5$ .

$y = - \frac{3}{10} \cdot (5 ((0) + 1) {((0) - 2)}^{2})$

Step 2.2.7.1.3

Cancel the common factor of $5$ .

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Step 2.2.7.1.3.1

Move the leading negative in $- \frac{3}{10}$ into the numerator.

$y = \frac{- 3}{10} \cdot (5 ((0) + 1) {((0) - 2)}^{2})$

Step 2.2.7.1.3.2

Factor $5$ out of $10$ .

$y = \frac{- 3}{5 (2)} \cdot (5 ((0) + 1) {((0) - 2)}^{2})$

Step 2.2.7.1.3.3

Factor $5$ out of $5 ((0) + 1) {((0) - 2)}^{2}$ .

$y = \frac{- 3}{5 (2)} \cdot (5 (((0) + 1) {((0) - 2)}^{2}))$

Step 2.2.7.1.3.4

Cancel the common factor.

$y = \frac{- 3}{5 \cdot 2} \cdot (5 (((0) + 1) {((0) - 2)}^{2}))$

Step 2.2.7.1.3.5

Rewrite the expression.

$y = \frac{- 3}{2} \cdot (((0) + 1) {((0) - 2)}^{2})$

Step 2.2.7.1.4

Combine ${((0) - 2)}^{2}$ and $\frac{- 3}{2}$ .

$y = \frac{{((0) - 2)}^{2} \cdot - 3}{2} \cdot ((0) + 1)$

Step 2.2.7.2

Simplify the numerator.

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Step 2.2.7.2.1

Subtract $2$ from $0$ .

$y = \frac{{(- 2)}^{2} \cdot - 3}{2} \cdot ((0) + 1)$

Step 2.2.7.2.2

Raise $- 2$ to the power of $2$ .

$y = \frac{4 \cdot - 3}{2} \cdot ((0) + 1)$

Step 2.2.7.3

Simplify the expression.

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Step 2.2.7.3.1

Multiply $4$ by $- 3$ .

$y = \frac{- 12}{2} \cdot ((0) + 1)$

Step 2.2.7.3.2

Divide $- 12$ by $2$ .

$y = - 6 \cdot ((0) + 1)$

Step 2.2.7.3.3

Add $0$ and $1$ .

$y = - 6 \cdot 1$

Step 2.2.7.3.4

Multiply $- 6$ by $1$ .

$y = - 6$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, - 6)$

Step 3

List the intersections.

x-intercept(s): $(2, 0), (- 1, 0), (- 5, 0)$

y-intercept(s): $(0, - 6)$

Step 4