Algebra Examples

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

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

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

Step 3

Simplify terms.

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Step 3.1

Combine the opposite terms in $x^{2} x^{2} + x^{2} (- x) + x^{2} \cdot 3 + x \cdot x^{2} + x (- x) + x \cdot 3 - 3 x^{2} - 3 (- x) - 3 \cdot 3$ .

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Step 3.1.1

Reorder the factors in the terms $x^{2} (- x)$ and $x \cdot x^{2}$ .

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

Step 3.1.2

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

$x^{2} x^{2} + x^{2} \cdot 3 + 0 + x (- x) + x \cdot 3 - 3 x^{2} - 3 (- x) - 3 \cdot 3$

Step 3.1.3

Add $x^{2} x^{2} + x^{2} \cdot 3$ and $0$ .

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

Step 3.1.4

Reorder the factors in the terms $x^{2} \cdot 3$ and $- 3 x^{2}$ .

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

Step 3.1.5

Subtract $3 x^{2}$ from $3 x^{2}$ .

$x^{2} x^{2} + x (- x) + x \cdot 3 + 0 - 3 (- x) - 3 \cdot 3$

Step 3.1.6

Add $x^{2} x^{2} + x (- x) + x \cdot 3$ and $0$ .

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

Step 3.2

Simplify each term.

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Step 3.2.1

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

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Step 3.2.1.1

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

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

Step 3.2.1.2

Add $2$ and $2$ .

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

Step 3.2.2

Rewrite using the commutative property of multiplication.

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

Step 3.2.3

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

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Step 3.2.3.1

Move $x$ .

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

Step 3.2.3.2

Multiply $x$ by $x$ .

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

Step 3.2.4

Move $3$ to the left of $x$ .

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

Step 3.2.5

Multiply $- 1$ by $- 3$ .

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

Step 3.2.6

Multiply $- 3$ by $3$ .

$x^{4} - x^{2} + 3 x + 3 x - 9$

Step 3.3

Add $3 x$ and $3 x$ .

$x^{4} - x^{2} + 6 x - 9$