Algebra Examples

$(3 x^{4} + 2 x) \cdot (2 x^{2} - 5 x + 34)$

Step 1

Expand $(3 x^{4} + 2 x) (2 x^{2} - 5 x + 34)$ by multiplying each term in the first expression by each term in the second expression.

$3 x^{4} (2 x^{2}) + 3 x^{4} (- 5 x) + 3 x^{4} \cdot 34 + 2 x (2 x^{2}) + 2 x (- 5 x) + 2 x \cdot 34$

Step 2

Simplify each term.

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Step 2.1

Rewrite using the commutative property of multiplication.

$3 \cdot 2 x^{4} x^{2} + 3 x^{4} (- 5 x) + 3 x^{4} \cdot 34 + 2 x (2 x^{2}) + 2 x (- 5 x) + 2 x \cdot 34$

Step 2.2

Multiply $x^{4}$ by $x^{2}$ by adding the exponents.

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Step 2.2.1

Move $x^{2}$ .

$3 \cdot 2 (x^{2} x^{4}) + 3 x^{4} (- 5 x) + 3 x^{4} \cdot 34 + 2 x (2 x^{2}) + 2 x (- 5 x) + 2 x \cdot 34$

Step 2.2.2

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

$3 \cdot 2 x^{2 + 4} + 3 x^{4} (- 5 x) + 3 x^{4} \cdot 34 + 2 x (2 x^{2}) + 2 x (- 5 x) + 2 x \cdot 34$

Step 2.2.3

Add $2$ and $4$ .

$3 \cdot 2 x^{6} + 3 x^{4} (- 5 x) + 3 x^{4} \cdot 34 + 2 x (2 x^{2}) + 2 x (- 5 x) + 2 x \cdot 34$

Step 2.3

Multiply $3$ by $2$ .

$6 x^{6} + 3 x^{4} (- 5 x) + 3 x^{4} \cdot 34 + 2 x (2 x^{2}) + 2 x (- 5 x) + 2 x \cdot 34$

Step 2.4

Rewrite using the commutative property of multiplication.

$6 x^{6} + 3 \cdot - 5 x^{4} x + 3 x^{4} \cdot 34 + 2 x (2 x^{2}) + 2 x (- 5 x) + 2 x \cdot 34$

Step 2.5

Multiply $x^{4}$ by $x$ by adding the exponents.

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Step 2.5.1

Move $x$ .

$6 x^{6} + 3 \cdot - 5 (x \cdot x^{4}) + 3 x^{4} \cdot 34 + 2 x (2 x^{2}) + 2 x (- 5 x) + 2 x \cdot 34$

Step 2.5.2

Multiply $x$ by $x^{4}$ .

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Step 2.5.2.1

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

$6 x^{6} + 3 \cdot - 5 (x^{1} x^{4}) + 3 x^{4} \cdot 34 + 2 x (2 x^{2}) + 2 x (- 5 x) + 2 x \cdot 34$

Step 2.5.2.2

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

$6 x^{6} + 3 \cdot - 5 x^{1 + 4} + 3 x^{4} \cdot 34 + 2 x (2 x^{2}) + 2 x (- 5 x) + 2 x \cdot 34$

Step 2.5.3

Add $1$ and $4$ .

$6 x^{6} + 3 \cdot - 5 x^{5} + 3 x^{4} \cdot 34 + 2 x (2 x^{2}) + 2 x (- 5 x) + 2 x \cdot 34$

Step 2.6

Multiply $3$ by $- 5$ .

$6 x^{6} - 15 x^{5} + 3 x^{4} \cdot 34 + 2 x (2 x^{2}) + 2 x (- 5 x) + 2 x \cdot 34$

Step 2.7

Multiply $34$ by $3$ .

$6 x^{6} - 15 x^{5} + 102 x^{4} + 2 x (2 x^{2}) + 2 x (- 5 x) + 2 x \cdot 34$

Step 2.8

Rewrite using the commutative property of multiplication.

$6 x^{6} - 15 x^{5} + 102 x^{4} + 2 \cdot 2 x \cdot x^{2} + 2 x (- 5 x) + 2 x \cdot 34$

Step 2.9

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

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Step 2.9.1

Move $x^{2}$ .

$6 x^{6} - 15 x^{5} + 102 x^{4} + 2 \cdot 2 (x^{2} x) + 2 x (- 5 x) + 2 x \cdot 34$

Step 2.9.2

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

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Step 2.9.2.1

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

$6 x^{6} - 15 x^{5} + 102 x^{4} + 2 \cdot 2 (x^{2} x^{1}) + 2 x (- 5 x) + 2 x \cdot 34$

Step 2.9.2.2

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

$6 x^{6} - 15 x^{5} + 102 x^{4} + 2 \cdot 2 x^{2 + 1} + 2 x (- 5 x) + 2 x \cdot 34$

Step 2.9.3

Add $2$ and $1$ .

$6 x^{6} - 15 x^{5} + 102 x^{4} + 2 \cdot 2 x^{3} + 2 x (- 5 x) + 2 x \cdot 34$

Step 2.10

Multiply $2$ by $2$ .

$6 x^{6} - 15 x^{5} + 102 x^{4} + 4 x^{3} + 2 x (- 5 x) + 2 x \cdot 34$

Step 2.11

Rewrite using the commutative property of multiplication.

$6 x^{6} - 15 x^{5} + 102 x^{4} + 4 x^{3} + 2 \cdot - 5 x \cdot x + 2 x \cdot 34$

Step 2.12

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

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Step 2.12.1

Move $x$ .

$6 x^{6} - 15 x^{5} + 102 x^{4} + 4 x^{3} + 2 \cdot - 5 (x \cdot x) + 2 x \cdot 34$

Step 2.12.2

Multiply $x$ by $x$ .

$6 x^{6} - 15 x^{5} + 102 x^{4} + 4 x^{3} + 2 \cdot - 5 x^{2} + 2 x \cdot 34$

Step 2.13

Multiply $2$ by $- 5$ .

$6 x^{6} - 15 x^{5} + 102 x^{4} + 4 x^{3} - 10 x^{2} + 2 x \cdot 34$

Step 2.14

Multiply $34$ by $2$ .

$6 x^{6} - 15 x^{5} + 102 x^{4} + 4 x^{3} - 10 x^{2} + 68 x$