Finite Math Examples

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

Step 1

Simplify and reorder the polynomial.

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Step 1.1

Rewrite ${(x^{2} + 3 x - 10)}^{2}$ as $(x^{2} + 3 x - 10) (x^{2} + 3 x - 10)$ .

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

Step 1.2

Expand $(x^{2} + 3 x - 10) (x^{2} + 3 x - 10)$ by multiplying each term in the first expression by each term in the second expression.

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

Step 1.3

Simplify terms.

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

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

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Step 1.3.1.1.1

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

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

Step 1.3.1.1.2

Add $2$ and $2$ .

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

Step 1.3.1.2

Rewrite using the commutative property of multiplication.

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

Step 1.3.1.3

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

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Step 1.3.1.3.1

Move $x$ .

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

Step 1.3.1.3.2

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

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Step 1.3.1.3.2.1

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

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

Step 1.3.1.3.2.2

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

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

Step 1.3.1.3.3

Add $1$ and $2$ .

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

Step 1.3.1.4

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

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

Step 1.3.1.5

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

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Step 1.3.1.5.1

Move $x^{2}$ .

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

Step 1.3.1.5.2

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

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Step 1.3.1.5.2.1

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

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

Step 1.3.1.5.2.2

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

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

Step 1.3.1.5.3

Add $2$ and $1$ .

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

Step 1.3.1.6

Rewrite using the commutative property of multiplication.

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

Step 1.3.1.7

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

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Step 1.3.1.7.1

Move $x$ .

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

Step 1.3.1.7.2

Multiply $x$ by $x$ .

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

Step 1.3.1.8

Multiply $3$ by $3$ .

$x^{4} + 3 x^{3} - 10 x^{2} + 3 x^{3} + 9 x^{2} + 3 x \cdot - 10 - 10 x^{2} - 10 (3 x) - 10 \cdot - 10$

Step 1.3.1.9

Multiply $- 10$ by $3$ .

$x^{4} + 3 x^{3} - 10 x^{2} + 3 x^{3} + 9 x^{2} - 30 x - 10 x^{2} - 10 (3 x) - 10 \cdot - 10$

Step 1.3.1.10

Multiply $3$ by $- 10$ .

$x^{4} + 3 x^{3} - 10 x^{2} + 3 x^{3} + 9 x^{2} - 30 x - 10 x^{2} - 30 x - 10 \cdot - 10$

Step 1.3.1.11

Multiply $- 10$ by $- 10$ .

$x^{4} + 3 x^{3} - 10 x^{2} + 3 x^{3} + 9 x^{2} - 30 x - 10 x^{2} - 30 x + 100$

Step 1.3.2

Simplify by adding terms.

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Step 1.3.2.1

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

$x^{4} + 6 x^{3} - 10 x^{2} + 9 x^{2} - 30 x - 10 x^{2} - 30 x + 100$

Step 1.3.2.2

Add $- 10 x^{2}$ and $9 x^{2}$ .

$x^{4} + 6 x^{3} - x^{2} - 30 x - 10 x^{2} - 30 x + 100$

Step 1.3.2.3

Subtract $10 x^{2}$ from $- x^{2}$ .

$x^{4} + 6 x^{3} - 11 x^{2} - 30 x - 30 x + 100$

Step 1.3.2.4

Subtract $30 x$ from $- 30 x$ .

$x^{4} + 6 x^{3} - 11 x^{2} - 60 x + 100$

Step 2

The largest exponent is the degree of the polynomial.

$4$