Algebra Examples

$(4 x^{5} + x^{3} - 7 x^{2} + 2) (3 x - 1)$

Step 1

Expand $(4 x^{5} + x^{3} - 7 x^{2} + 2) (3 x - 1)$ by multiplying each term in the first expression by each term in the second expression.

$4 x^{5} (3 x) + 4 x^{5} \cdot - 1 + x^{3} (3 x) + x^{3} \cdot - 1 - 7 x^{2} (3 x) - 7 x^{2} \cdot - 1 + 2 (3 x) + 2 \cdot - 1$

Step 2

Simplify terms.

Tap for more steps...

Step 2.1

Simplify each term.

Tap for more steps...

Step 2.1.1

Rewrite using the commutative property of multiplication.

$4 \cdot 3 x^{5} x + 4 x^{5} \cdot - 1 + x^{3} (3 x) + x^{3} \cdot - 1 - 7 x^{2} (3 x) - 7 x^{2} \cdot - 1 + 2 (3 x) + 2 \cdot - 1$

Step 2.1.2

Multiply $x^{5}$ by $x$ by adding the exponents.

Tap for more steps...

Step 2.1.2.1

Move $x$ .

$4 \cdot 3 (x \cdot x^{5}) + 4 x^{5} \cdot - 1 + x^{3} (3 x) + x^{3} \cdot - 1 - 7 x^{2} (3 x) - 7 x^{2} \cdot - 1 + 2 (3 x) + 2 \cdot - 1$

Step 2.1.2.2

Multiply $x$ by $x^{5}$ .

Tap for more steps...

Step 2.1.2.2.1

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

$4 \cdot 3 (x^{1} x^{5}) + 4 x^{5} \cdot - 1 + x^{3} (3 x) + x^{3} \cdot - 1 - 7 x^{2} (3 x) - 7 x^{2} \cdot - 1 + 2 (3 x) + 2 \cdot - 1$

Step 2.1.2.2.2

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

$4 \cdot 3 x^{1 + 5} + 4 x^{5} \cdot - 1 + x^{3} (3 x) + x^{3} \cdot - 1 - 7 x^{2} (3 x) - 7 x^{2} \cdot - 1 + 2 (3 x) + 2 \cdot - 1$

Step 2.1.2.3

Add $1$ and $5$ .

$4 \cdot 3 x^{6} + 4 x^{5} \cdot - 1 + x^{3} (3 x) + x^{3} \cdot - 1 - 7 x^{2} (3 x) - 7 x^{2} \cdot - 1 + 2 (3 x) + 2 \cdot - 1$

Step 2.1.3

Multiply $4$ by $3$ .

$12 x^{6} + 4 x^{5} \cdot - 1 + x^{3} (3 x) + x^{3} \cdot - 1 - 7 x^{2} (3 x) - 7 x^{2} \cdot - 1 + 2 (3 x) + 2 \cdot - 1$

Step 2.1.4

Multiply $- 1$ by $4$ .

$12 x^{6} - 4 x^{5} + x^{3} (3 x) + x^{3} \cdot - 1 - 7 x^{2} (3 x) - 7 x^{2} \cdot - 1 + 2 (3 x) + 2 \cdot - 1$

Step 2.1.5

Rewrite using the commutative property of multiplication.

$12 x^{6} - 4 x^{5} + 3 x^{3} x + x^{3} \cdot - 1 - 7 x^{2} (3 x) - 7 x^{2} \cdot - 1 + 2 (3 x) + 2 \cdot - 1$

Step 2.1.6

Multiply $x^{3}$ by $x$ by adding the exponents.

Tap for more steps...

Step 2.1.6.1

Move $x$ .

$12 x^{6} - 4 x^{5} + 3 (x \cdot x^{3}) + x^{3} \cdot - 1 - 7 x^{2} (3 x) - 7 x^{2} \cdot - 1 + 2 (3 x) + 2 \cdot - 1$

Step 2.1.6.2

Multiply $x$ by $x^{3}$ .

Tap for more steps...

Step 2.1.6.2.1

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

$12 x^{6} - 4 x^{5} + 3 (x^{1} x^{3}) + x^{3} \cdot - 1 - 7 x^{2} (3 x) - 7 x^{2} \cdot - 1 + 2 (3 x) + 2 \cdot - 1$

Step 2.1.6.2.2

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

$12 x^{6} - 4 x^{5} + 3 x^{1 + 3} + x^{3} \cdot - 1 - 7 x^{2} (3 x) - 7 x^{2} \cdot - 1 + 2 (3 x) + 2 \cdot - 1$

Step 2.1.6.3

Add $1$ and $3$ .

$12 x^{6} - 4 x^{5} + 3 x^{4} + x^{3} \cdot - 1 - 7 x^{2} (3 x) - 7 x^{2} \cdot - 1 + 2 (3 x) + 2 \cdot - 1$

Step 2.1.7

Move $- 1$ to the left of $x^{3}$ .

$12 x^{6} - 4 x^{5} + 3 x^{4} - 1 \cdot x^{3} - 7 x^{2} (3 x) - 7 x^{2} \cdot - 1 + 2 (3 x) + 2 \cdot - 1$

Step 2.1.8

Rewrite $- 1 x^{3}$ as $- x^{3}$ .

$12 x^{6} - 4 x^{5} + 3 x^{4} - x^{3} - 7 x^{2} (3 x) - 7 x^{2} \cdot - 1 + 2 (3 x) + 2 \cdot - 1$

Step 2.1.9

Rewrite using the commutative property of multiplication.

$12 x^{6} - 4 x^{5} + 3 x^{4} - x^{3} - 7 \cdot 3 x^{2} x - 7 x^{2} \cdot - 1 + 2 (3 x) + 2 \cdot - 1$

Step 2.1.10

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

Tap for more steps...

Step 2.1.10.1

Move $x$ .

$12 x^{6} - 4 x^{5} + 3 x^{4} - x^{3} - 7 \cdot 3 (x \cdot x^{2}) - 7 x^{2} \cdot - 1 + 2 (3 x) + 2 \cdot - 1$

Step 2.1.10.2

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

Tap for more steps...

Step 2.1.10.2.1

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

$12 x^{6} - 4 x^{5} + 3 x^{4} - x^{3} - 7 \cdot 3 (x^{1} x^{2}) - 7 x^{2} \cdot - 1 + 2 (3 x) + 2 \cdot - 1$

Step 2.1.10.2.2

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

$12 x^{6} - 4 x^{5} + 3 x^{4} - x^{3} - 7 \cdot 3 x^{1 + 2} - 7 x^{2} \cdot - 1 + 2 (3 x) + 2 \cdot - 1$

Step 2.1.10.3

Add $1$ and $2$ .

$12 x^{6} - 4 x^{5} + 3 x^{4} - x^{3} - 7 \cdot 3 x^{3} - 7 x^{2} \cdot - 1 + 2 (3 x) + 2 \cdot - 1$

Step 2.1.11

Multiply $- 7$ by $3$ .

$12 x^{6} - 4 x^{5} + 3 x^{4} - x^{3} - 21 x^{3} - 7 x^{2} \cdot - 1 + 2 (3 x) + 2 \cdot - 1$

Step 2.1.12

Multiply $- 1$ by $- 7$ .

$12 x^{6} - 4 x^{5} + 3 x^{4} - x^{3} - 21 x^{3} + 7 x^{2} + 2 (3 x) + 2 \cdot - 1$

Step 2.1.13

Multiply $3$ by $2$ .

$12 x^{6} - 4 x^{5} + 3 x^{4} - x^{3} - 21 x^{3} + 7 x^{2} + 6 x + 2 \cdot - 1$

Step 2.1.14

Multiply $2$ by $- 1$ .

$12 x^{6} - 4 x^{5} + 3 x^{4} - x^{3} - 21 x^{3} + 7 x^{2} + 6 x - 2$

Step 2.2

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

$12 x^{6} - 4 x^{5} + 3 x^{4} - 22 x^{3} + 7 x^{2} + 6 x - 2$