Algebra Examples

${(x - 2)}^{2} (x + 1) + 3 (x - 2) (x + 1) = (x - 2) P$

Step 1

Rewrite the equation as $(x - 2) P = {(x - 2)}^{2} (x + 1) + 3 (x - 2) (x + 1)$ .

$(x - 2) P = {(x - 2)}^{2} (x + 1) + 3 (x - 2) (x + 1)$

Step 2

Divide each term in $(x - 2) P = {(x - 2)}^{2} (x + 1) + 3 (x - 2) (x + 1)$ by $x - 2$ and simplify.

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Step 2.1

Divide each term in $(x - 2) P = {(x - 2)}^{2} (x + 1) + 3 (x - 2) (x + 1)$ by $x - 2$ .

$\frac{(x - 2) P}{x - 2} = \frac{{(x - 2)}^{2} (x + 1)}{x - 2} + \frac{3 (x - 2) (x + 1)}{x - 2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $x - 2$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{(x - 2) P}{x - 2} = \frac{{(x - 2)}^{2} (x + 1)}{x - 2} + \frac{3 (x - 2) (x + 1)}{x - 2}$

Step 2.2.1.2

Divide $P$ by $1$ .

$P = \frac{{(x - 2)}^{2} (x + 1)}{x - 2} + \frac{3 (x - 2) (x + 1)}{x - 2}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Combine the numerators over the common denominator.

$P = \frac{{(x - 2)}^{2} (x + 1) + 3 (x - 2) (x + 1)}{x - 2}$

Step 2.3.2

Simplify each term.

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Step 2.3.2.1

Apply the distributive property.

$P = \frac{{(x - 2)}^{2} x + {(x - 2)}^{2} \cdot 1 + 3 (x - 2) (x + 1)}{x - 2}$

Step 2.3.2.2

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

$P = \frac{{(x - 2)}^{2} x + {(x - 2)}^{2} + 3 (x - 2) (x + 1)}{x - 2}$

Step 2.3.2.3

Apply the distributive property.

$P = \frac{{(x - 2)}^{2} x + {(x - 2)}^{2} + (3 x + 3 \cdot - 2) (x + 1)}{x - 2}$

Step 2.3.2.4

Multiply $3$ by $- 2$ .

$P = \frac{{(x - 2)}^{2} x + {(x - 2)}^{2} + (3 x - 6) (x + 1)}{x - 2}$

Step 2.3.2.5

Expand $(3 x - 6) (x + 1)$ using the FOIL Method.

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Step 2.3.2.5.1

Apply the distributive property.

$P = \frac{{(x - 2)}^{2} x + {(x - 2)}^{2} + 3 x (x + 1) - 6 (x + 1)}{x - 2}$

Step 2.3.2.5.2

Apply the distributive property.

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

Step 2.3.2.5.3

Apply the distributive property.

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

Step 2.3.2.6

Simplify and combine like terms.

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Step 2.3.2.6.1

Simplify each term.

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Step 2.3.2.6.1.1

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

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Step 2.3.2.6.1.1.1

Move $x$ .

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

Step 2.3.2.6.1.1.2

Multiply $x$ by $x$ .

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

Step 2.3.2.6.1.2

Multiply $3$ by $1$ .

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

Step 2.3.2.6.1.3

Multiply $- 6$ by $1$ .

$P = \frac{{(x - 2)}^{2} x + {(x - 2)}^{2} + 3 x^{2} + 3 x - 6 x - 6}{x - 2}$

Step 2.3.2.6.2

Subtract $6 x$ from $3 x$ .

$P = \frac{{(x - 2)}^{2} x + {(x - 2)}^{2} + 3 x^{2} - 3 x - 6}{x - 2}$

Step 2.3.3

Simplify the expression.

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Step 2.3.3.1

Reorder factors in ${(x - 2)}^{2} x + {(x - 2)}^{2} + 3 x^{2} - 3 x - 6$ .

$P = \frac{x {(x - 2)}^{2} + {(x - 2)}^{2} + 3 x^{2} - 3 x - 6}{x - 2}$

Step 2.3.3.2

Reorder terms.

$P = \frac{3 x^{2} + {(x - 2)}^{2} + x {(x - 2)}^{2} - 3 x - 6}{x - 2}$