Algebra Examples

$(x^{2} + 2 x - 2) (x^{2} + x + 10)$

Step 1

To write a polynomial in standard form, simplify and then arrange the terms in descending order.

$a x^{2} + b x + c$

Step 2

Expand $(x^{2} + 2 x - 2) (x^{2} + x + 10)$ by multiplying each term in the first expression by each term in the second expression.

$x^{2} x^{2} + x^{2} x + x^{2} \cdot 10 + 2 x \cdot x^{2} + 2 x \cdot x + 2 x \cdot 10 - 2 x^{2} - 2 x - 2 \cdot 10$

Step 3

Simplify terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

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

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Step 3.1.1.1

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

$x^{2 + 2} + x^{2} x + x^{2} \cdot 10 + 2 x \cdot x^{2} + 2 x \cdot x + 2 x \cdot 10 - 2 x^{2} - 2 x - 2 \cdot 10$

Step 3.1.1.2

Add $2$ and $2$ .

$x^{4} + x^{2} x + x^{2} \cdot 10 + 2 x \cdot x^{2} + 2 x \cdot x + 2 x \cdot 10 - 2 x^{2} - 2 x - 2 \cdot 10$

Step 3.1.2

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

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Step 3.1.2.1

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

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Step 3.1.2.1.1

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

$x^{4} + x^{2} x^{1} + x^{2} \cdot 10 + 2 x \cdot x^{2} + 2 x \cdot x + 2 x \cdot 10 - 2 x^{2} - 2 x - 2 \cdot 10$

Step 3.1.2.1.2

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

$x^{4} + x^{2 + 1} + x^{2} \cdot 10 + 2 x \cdot x^{2} + 2 x \cdot x + 2 x \cdot 10 - 2 x^{2} - 2 x - 2 \cdot 10$

Step 3.1.2.2

Add $2$ and $1$ .

$x^{4} + x^{3} + x^{2} \cdot 10 + 2 x \cdot x^{2} + 2 x \cdot x + 2 x \cdot 10 - 2 x^{2} - 2 x - 2 \cdot 10$

Step 3.1.3

Move $10$ to the left of $x^{2}$ .

$x^{4} + x^{3} + 10 \cdot x^{2} + 2 x \cdot x^{2} + 2 x \cdot x + 2 x \cdot 10 - 2 x^{2} - 2 x - 2 \cdot 10$

Step 3.1.4

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

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Step 3.1.4.1

Move $x^{2}$ .

$x^{4} + x^{3} + 10 x^{2} + 2 (x^{2} x) + 2 x \cdot x + 2 x \cdot 10 - 2 x^{2} - 2 x - 2 \cdot 10$

Step 3.1.4.2

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

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Step 3.1.4.2.1

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

$x^{4} + x^{3} + 10 x^{2} + 2 (x^{2} x^{1}) + 2 x \cdot x + 2 x \cdot 10 - 2 x^{2} - 2 x - 2 \cdot 10$

Step 3.1.4.2.2

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

$x^{4} + x^{3} + 10 x^{2} + 2 x^{2 + 1} + 2 x \cdot x + 2 x \cdot 10 - 2 x^{2} - 2 x - 2 \cdot 10$

Step 3.1.4.3

Add $2$ and $1$ .

$x^{4} + x^{3} + 10 x^{2} + 2 x^{3} + 2 x \cdot x + 2 x \cdot 10 - 2 x^{2} - 2 x - 2 \cdot 10$

Step 3.1.5

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

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Step 3.1.5.1

Move $x$ .

$x^{4} + x^{3} + 10 x^{2} + 2 x^{3} + 2 (x \cdot x) + 2 x \cdot 10 - 2 x^{2} - 2 x - 2 \cdot 10$

Step 3.1.5.2

Multiply $x$ by $x$ .

$x^{4} + x^{3} + 10 x^{2} + 2 x^{3} + 2 x^{2} + 2 x \cdot 10 - 2 x^{2} - 2 x - 2 \cdot 10$

Step 3.1.6

Multiply $10$ by $2$ .

$x^{4} + x^{3} + 10 x^{2} + 2 x^{3} + 2 x^{2} + 20 x - 2 x^{2} - 2 x - 2 \cdot 10$

Step 3.1.7

Multiply $- 2$ by $10$ .

$x^{4} + x^{3} + 10 x^{2} + 2 x^{3} + 2 x^{2} + 20 x - 2 x^{2} - 2 x - 20$

Step 3.2

Simplify by adding terms.

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Step 3.2.1

Combine the opposite terms in $x^{4} + x^{3} + 10 x^{2} + 2 x^{3} + 2 x^{2} + 20 x - 2 x^{2} - 2 x - 20$ .

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Step 3.2.1.1

Subtract $2 x^{2}$ from $2 x^{2}$ .

$x^{4} + x^{3} + 10 x^{2} + 2 x^{3} + 20 x + 0 - 2 x - 20$

Step 3.2.1.2

Add $x^{4} + x^{3} + 10 x^{2} + 2 x^{3} + 20 x$ and $0$ .

$x^{4} + x^{3} + 10 x^{2} + 2 x^{3} + 20 x - 2 x - 20$

Step 3.2.2

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

$x^{4} + 3 x^{3} + 10 x^{2} + 20 x - 2 x - 20$

Step 3.2.3

Subtract $2 x$ from $20 x$ .

$x^{4} + 3 x^{3} + 10 x^{2} + 18 x - 20$