Algebra Examples

$3 x (x - 7) {(x - 1)}^{2}$

Step 1

Apply the distributive property.

$(3 x \cdot x + 3 x \cdot - 7) {(x - 1)}^{2}$

Step 2

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

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Step 2.1

Move $x$ .

$(3 (x \cdot x) + 3 x \cdot - 7) {(x - 1)}^{2}$

Step 2.2

Multiply $x$ by $x$ .

$(3 x^{2} + 3 x \cdot - 7) {(x - 1)}^{2}$

Step 3

Simplify the expression.

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Step 3.1

Multiply $- 7$ by $3$ .

$(3 x^{2} - 21 x) {(x - 1)}^{2}$

Step 3.2

Rewrite ${(x - 1)}^{2}$ as $(x - 1) (x - 1)$ .

$(3 x^{2} - 21 x) ((x - 1) (x - 1))$

Step 4

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

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Step 4.1

Apply the distributive property.

$(3 x^{2} - 21 x) (x (x - 1) - 1 (x - 1))$

Step 4.2

Apply the distributive property.

$(3 x^{2} - 21 x) (x \cdot x + x \cdot - 1 - 1 (x - 1))$

Step 4.3

Apply the distributive property.

$(3 x^{2} - 21 x) (x \cdot x + x \cdot - 1 - 1 x - 1 \cdot - 1)$

Step 5

Simplify and combine like terms.

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Step 5.1

Simplify each term.

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Step 5.1.1

Multiply $x$ by $x$ .

$(3 x^{2} - 21 x) (x^{2} + x \cdot - 1 - 1 x - 1 \cdot - 1)$

Step 5.1.2

Move $- 1$ to the left of $x$ .

$(3 x^{2} - 21 x) (x^{2} - 1 \cdot x - 1 x - 1 \cdot - 1)$

Step 5.1.3

Rewrite $- 1 x$ as $- x$ .

$(3 x^{2} - 21 x) (x^{2} - x - 1 x - 1 \cdot - 1)$

Step 5.1.4

Rewrite $- 1 x$ as $- x$ .

$(3 x^{2} - 21 x) (x^{2} - x - x - 1 \cdot - 1)$

Step 5.1.5

Multiply $- 1$ by $- 1$ .

$(3 x^{2} - 21 x) (x^{2} - x - x + 1)$

Step 5.2

Subtract $x$ from $- x$ .

$(3 x^{2} - 21 x) (x^{2} - 2 x + 1)$

Step 6

Expand $(3 x^{2} - 21 x) (x^{2} - 2 x + 1)$ by multiplying each term in the first expression by each term in the second expression.

$3 x^{2} x^{2} + 3 x^{2} (- 2 x) + 3 x^{2} \cdot 1 - 21 x \cdot x^{2} - 21 x (- 2 x) - 21 x \cdot 1$

Step 7

Simplify terms.

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Step 7.1

Simplify each term.

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Step 7.1.1

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

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Step 7.1.1.1

Move $x^{2}$ .

$3 (x^{2} x^{2}) + 3 x^{2} (- 2 x) + 3 x^{2} \cdot 1 - 21 x \cdot x^{2} - 21 x (- 2 x) - 21 x \cdot 1$

Step 7.1.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$3 x^{2 + 2} + 3 x^{2} (- 2 x) + 3 x^{2} \cdot 1 - 21 x \cdot x^{2} - 21 x (- 2 x) - 21 x \cdot 1$

Step 7.1.1.3

Add $2$ and $2$ .

$3 x^{4} + 3 x^{2} (- 2 x) + 3 x^{2} \cdot 1 - 21 x \cdot x^{2} - 21 x (- 2 x) - 21 x \cdot 1$

Step 7.1.2

Rewrite using the commutative property of multiplication.

$3 x^{4} + 3 \cdot - 2 x^{2} x + 3 x^{2} \cdot 1 - 21 x \cdot x^{2} - 21 x (- 2 x) - 21 x \cdot 1$

Step 7.1.3

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 7.1.3.1

Move $x$ .

$3 x^{4} + 3 \cdot - 2 (x \cdot x^{2}) + 3 x^{2} \cdot 1 - 21 x \cdot x^{2} - 21 x (- 2 x) - 21 x \cdot 1$

Step 7.1.3.2

Multiply $x$ by $x^{2}$ .

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Step 7.1.3.2.1

Raise $x$ to the power of $1$ .

$3 x^{4} + 3 \cdot - 2 (x^{1} x^{2}) + 3 x^{2} \cdot 1 - 21 x \cdot x^{2} - 21 x (- 2 x) - 21 x \cdot 1$

Step 7.1.3.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$3 x^{4} + 3 \cdot - 2 x^{1 + 2} + 3 x^{2} \cdot 1 - 21 x \cdot x^{2} - 21 x (- 2 x) - 21 x \cdot 1$

Step 7.1.3.3

Add $1$ and $2$ .

$3 x^{4} + 3 \cdot - 2 x^{3} + 3 x^{2} \cdot 1 - 21 x \cdot x^{2} - 21 x (- 2 x) - 21 x \cdot 1$

Step 7.1.4

Multiply $3$ by $- 2$ .

$3 x^{4} - 6 x^{3} + 3 x^{2} \cdot 1 - 21 x \cdot x^{2} - 21 x (- 2 x) - 21 x \cdot 1$

Step 7.1.5

Multiply $3$ by $1$ .

$3 x^{4} - 6 x^{3} + 3 x^{2} - 21 x \cdot x^{2} - 21 x (- 2 x) - 21 x \cdot 1$

Step 7.1.6

Multiply $x$ by $x^{2}$ by adding the exponents.

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Step 7.1.6.1

Move $x^{2}$ .

$3 x^{4} - 6 x^{3} + 3 x^{2} - 21 (x^{2} x) - 21 x (- 2 x) - 21 x \cdot 1$

Step 7.1.6.2

Multiply $x^{2}$ by $x$ .

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Step 7.1.6.2.1

Raise $x$ to the power of $1$ .

$3 x^{4} - 6 x^{3} + 3 x^{2} - 21 (x^{2} x^{1}) - 21 x (- 2 x) - 21 x \cdot 1$

Step 7.1.6.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$3 x^{4} - 6 x^{3} + 3 x^{2} - 21 x^{2 + 1} - 21 x (- 2 x) - 21 x \cdot 1$

Step 7.1.6.3

Add $2$ and $1$ .

$3 x^{4} - 6 x^{3} + 3 x^{2} - 21 x^{3} - 21 x (- 2 x) - 21 x \cdot 1$

Step 7.1.7

Rewrite using the commutative property of multiplication.

$3 x^{4} - 6 x^{3} + 3 x^{2} - 21 x^{3} - 21 \cdot - 2 x \cdot x - 21 x \cdot 1$

Step 7.1.8

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

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Step 7.1.8.1

Move $x$ .

$3 x^{4} - 6 x^{3} + 3 x^{2} - 21 x^{3} - 21 \cdot - 2 (x \cdot x) - 21 x \cdot 1$

Step 7.1.8.2

Multiply $x$ by $x$ .

$3 x^{4} - 6 x^{3} + 3 x^{2} - 21 x^{3} - 21 \cdot - 2 x^{2} - 21 x \cdot 1$

Step 7.1.9

Multiply $- 21$ by $- 2$ .

$3 x^{4} - 6 x^{3} + 3 x^{2} - 21 x^{3} + 42 x^{2} - 21 x \cdot 1$

Step 7.1.10

Multiply $- 21$ by $1$ .

$3 x^{4} - 6 x^{3} + 3 x^{2} - 21 x^{3} + 42 x^{2} - 21 x$

Step 7.2

Simplify by adding terms.

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Step 7.2.1

Subtract $21 x^{3}$ from $- 6 x^{3}$ .

$3 x^{4} - 27 x^{3} + 3 x^{2} + 42 x^{2} - 21 x$

Step 7.2.2

Add $3 x^{2}$ and $42 x^{2}$ .

$3 x^{4} - 27 x^{3} + 45 x^{2} - 21 x$