Solve x/ (x-1)(x-2)(x-3)

Question

Simplify the expression

\frac{x ^{3} - 5 x ^{2} + 6 x}{x - 1}

Evaluate

\frac{x}{x - 1} \times (x - 2) (x - 3)

Multiply the first two terms

\frac{x ( x - 2 )}{x - 1} \times (x - 3)

Multiply the terms

\frac{x ( x - 2 ) ( x - 3 )}{x - 1}

Solution

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Evaluate

x (x - 2) (x - 3)

Multiply the terms

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Evaluate

x (x - 2)

Apply the distributive property

x \times x - x \times 2

Multiply the terms

x^{2} - x \times 2

Use the commutative property to reorder the terms

x^{2} - 2 x

(x^{2} - 2 x) (x - 3)

Apply the distributive property

x^{2} \times x - x^{2} \times 3 - 2 x \times x - (- 2 x \times 3)

Multiply the terms

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Evaluate

x^{2} \times x

Use the product rule a^{n} \times a^{m} = a^{n + m} to simplify the expression

x^{2 + 1}

Add the numbers

x^{3}

x^{3} - x^{2} \times 3 - 2 x \times x - (- 2 x \times 3)

Use the commutative property to reorder the terms

x^{3} - 3 x^{2} - 2 x \times x - (- 2 x \times 3)

Multiply the terms

x^{3} - 3 x^{2} - 2 x^{2} - (- 2 x \times 3)

Multiply the numbers

x^{3} - 3 x^{2} - 2 x^{2} - (- 6 x)

If a negative sign or a subtraction symbol appears outside parentheses, remove the parentheses and change the sign of every term within the parentheses

x^{3} - 3 x^{2} - 2 x^{2} + 6 x

Subtract the terms

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Evaluate

- 3 x^{2} - 2 x^{2}

Collect like terms by calculating the sum or difference of their coefficients

(- 3 - 2) x^{2}

Subtract the numbers

- 5 x^{2}

x^{3} - 5 x^{2} + 6 x

\frac{x ^{3} - 5 x ^{2} + 6 x}{x - 1}

Show Solution

Find the excluded values

x = 1

Evaluate

\frac{x}{x - 1} \times (x - 2) (x - 3)

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

x - 1 = 0

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

x = 0 + 1

Solution

x = 1

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Find the roots

x_{1} = 0, x_{2} = 2, x_{3} = 3

Evaluate

\frac{x}{x - 1} \times (x - 2) (x - 3)

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

\frac{x}{x - 1} \times (x - 2) (x - 3) = 0

Find the domain

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Evaluate

x - 1 \neq = 0

Move the constant to the right side

x \neq = 0 + 1

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

x \neq = 1

\frac{x}{x - 1} \times (x - 2) (x - 3) = 0, x \neq = 1

Calculate

\frac{x}{x - 1} \times (x - 2) (x - 3) = 0

Multiply the terms

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Multiply the terms

\frac{x}{x - 1} \times (x - 2) (x - 3)

Multiply the first two terms

\frac{x ( x - 2 )}{x - 1} \times (x - 3)

Multiply the terms

\frac{x ( x - 2 ) ( x - 3 )}{x - 1}

\frac{x ( x - 2 ) ( x - 3 )}{x - 1} = 0

Cross multiply

x (x - 2) (x - 3) = (x - 1) \times 0

Simplify the equation

x (x - 2) (x - 3) = 0

Separate the equation into 3 possible cases

x = 0 x - 2 = 0 x - 3 = 0

Solve the equation

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Evaluate

x - 2 = 0

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

x = 0 + 2

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

x = 2

x = 0 x = 2 x - 3 = 0

Solve the equation

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Evaluate

x - 3 = 0

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

x = 0 + 3

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

x = 3

x = 0 x = 2 x = 3

Check if the solution is in the defined range

x = 0 x = 2 x = 3, x \neq = 1

Find the intersection of the solution and the defined range

x = 0 x = 2 x = 3

Solution

x_{1} = 0, x_{2} = 2, x_{3} = 3

Show Solution