Algebra Examples

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

Step 1

Simplify each term.

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Step 1.1

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

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

Step 1.2

Simplify each term.

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Step 1.2.1

Rewrite using the commutative property of multiplication.

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

Step 1.2.2

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

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Step 1.2.2.1

Move $x^{2}$ .

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

Step 1.2.2.2

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

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Step 1.2.2.2.1

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

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

Step 1.2.2.2.2

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

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

Step 1.2.2.3

Add $2$ and $1$ .

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

Step 1.2.3

Rewrite using the commutative property of multiplication.

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

Step 1.2.4

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

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Step 1.2.4.1

Move $x$ .

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

Step 1.2.4.2

Multiply $x$ by $x$ .

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

Step 1.2.5

Move $- 4$ to the left of $x$ .

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

Step 1.2.6

Multiply $5$ by $7$ .

$5 x^{3} - 7 x^{2} - 4 x + 35 x^{2} + 7 (- 7 x) + 7 \cdot - 4 - (x - 3) (4 x^{3} - x^{2} + 4 x - 3)$

Step 1.2.7

Multiply $- 7$ by $7$ .

$5 x^{3} - 7 x^{2} - 4 x + 35 x^{2} - 49 x + 7 \cdot - 4 - (x - 3) (4 x^{3} - x^{2} + 4 x - 3)$

Step 1.2.8

Multiply $7$ by $- 4$ .

$5 x^{3} - 7 x^{2} - 4 x + 35 x^{2} - 49 x - 28 - (x - 3) (4 x^{3} - x^{2} + 4 x - 3)$

Step 1.3

Add $- 7 x^{2}$ and $35 x^{2}$ .

$5 x^{3} + 28 x^{2} - 4 x - 49 x - 28 - (x - 3) (4 x^{3} - x^{2} + 4 x - 3)$

Step 1.4

Subtract $49 x$ from $- 4 x$ .

$5 x^{3} + 28 x^{2} - 53 x - 28 - (x - 3) (4 x^{3} - x^{2} + 4 x - 3)$

Step 1.5

Apply the distributive property.

$5 x^{3} + 28 x^{2} - 53 x - 28 + (- x - - 3) (4 x^{3} - x^{2} + 4 x - 3)$

Step 1.6

Multiply $- 1$ by $- 3$ .

$5 x^{3} + 28 x^{2} - 53 x - 28 + (- x + 3) (4 x^{3} - x^{2} + 4 x - 3)$

Step 1.7

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

$5 x^{3} + 28 x^{2} - 53 x - 28 - x (4 x^{3}) - x (- x^{2}) - x (4 x) - x \cdot - 3 + 3 (4 x^{3}) + 3 (- x^{2}) + 3 (4 x) + 3 \cdot - 3$

Step 1.8

Simplify each term.

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Step 1.8.1

Rewrite using the commutative property of multiplication.

$5 x^{3} + 28 x^{2} - 53 x - 28 - 1 \cdot 4 x \cdot x^{3} - x (- x^{2}) - x (4 x) - x \cdot - 3 + 3 (4 x^{3}) + 3 (- x^{2}) + 3 (4 x) + 3 \cdot - 3$

Step 1.8.2

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

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Step 1.8.2.1

Move $x^{3}$ .

$5 x^{3} + 28 x^{2} - 53 x - 28 - 1 \cdot 4 (x^{3} x) - x (- x^{2}) - x (4 x) - x \cdot - 3 + 3 (4 x^{3}) + 3 (- x^{2}) + 3 (4 x) + 3 \cdot - 3$

Step 1.8.2.2

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

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Step 1.8.2.2.1

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

$5 x^{3} + 28 x^{2} - 53 x - 28 - 1 \cdot 4 (x^{3} x^{1}) - x (- x^{2}) - x (4 x) - x \cdot - 3 + 3 (4 x^{3}) + 3 (- x^{2}) + 3 (4 x) + 3 \cdot - 3$

Step 1.8.2.2.2

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

$5 x^{3} + 28 x^{2} - 53 x - 28 - 1 \cdot 4 x^{3 + 1} - x (- x^{2}) - x (4 x) - x \cdot - 3 + 3 (4 x^{3}) + 3 (- x^{2}) + 3 (4 x) + 3 \cdot - 3$

Step 1.8.2.3

Add $3$ and $1$ .

$5 x^{3} + 28 x^{2} - 53 x - 28 - 1 \cdot 4 x^{4} - x (- x^{2}) - x (4 x) - x \cdot - 3 + 3 (4 x^{3}) + 3 (- x^{2}) + 3 (4 x) + 3 \cdot - 3$

Step 1.8.3

Multiply $- 1$ by $4$ .

$5 x^{3} + 28 x^{2} - 53 x - 28 - 4 x^{4} - x (- x^{2}) - x (4 x) - x \cdot - 3 + 3 (4 x^{3}) + 3 (- x^{2}) + 3 (4 x) + 3 \cdot - 3$

Step 1.8.4

Rewrite using the commutative property of multiplication.

$5 x^{3} + 28 x^{2} - 53 x - 28 - 4 x^{4} - 1 \cdot - 1 x \cdot x^{2} - x (4 x) - x \cdot - 3 + 3 (4 x^{3}) + 3 (- x^{2}) + 3 (4 x) + 3 \cdot - 3$

Step 1.8.5

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

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Step 1.8.5.1

Move $x^{2}$ .

$5 x^{3} + 28 x^{2} - 53 x - 28 - 4 x^{4} - 1 \cdot - 1 (x^{2} x) - x (4 x) - x \cdot - 3 + 3 (4 x^{3}) + 3 (- x^{2}) + 3 (4 x) + 3 \cdot - 3$

Step 1.8.5.2

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

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Step 1.8.5.2.1

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

$5 x^{3} + 28 x^{2} - 53 x - 28 - 4 x^{4} - 1 \cdot - 1 (x^{2} x^{1}) - x (4 x) - x \cdot - 3 + 3 (4 x^{3}) + 3 (- x^{2}) + 3 (4 x) + 3 \cdot - 3$

Step 1.8.5.2.2

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

$5 x^{3} + 28 x^{2} - 53 x - 28 - 4 x^{4} - 1 \cdot - 1 x^{2 + 1} - x (4 x) - x \cdot - 3 + 3 (4 x^{3}) + 3 (- x^{2}) + 3 (4 x) + 3 \cdot - 3$

Step 1.8.5.3

Add $2$ and $1$ .

$5 x^{3} + 28 x^{2} - 53 x - 28 - 4 x^{4} - 1 \cdot - 1 x^{3} - x (4 x) - x \cdot - 3 + 3 (4 x^{3}) + 3 (- x^{2}) + 3 (4 x) + 3 \cdot - 3$

Step 1.8.6

Multiply $- 1$ by $- 1$ .

$5 x^{3} + 28 x^{2} - 53 x - 28 - 4 x^{4} + 1 x^{3} - x (4 x) - x \cdot - 3 + 3 (4 x^{3}) + 3 (- x^{2}) + 3 (4 x) + 3 \cdot - 3$

Step 1.8.7

Multiply $x^{3}$ by $1$ .

$5 x^{3} + 28 x^{2} - 53 x - 28 - 4 x^{4} + x^{3} - x (4 x) - x \cdot - 3 + 3 (4 x^{3}) + 3 (- x^{2}) + 3 (4 x) + 3 \cdot - 3$

Step 1.8.8

Rewrite using the commutative property of multiplication.

$5 x^{3} + 28 x^{2} - 53 x - 28 - 4 x^{4} + x^{3} - 1 \cdot 4 x \cdot x - x \cdot - 3 + 3 (4 x^{3}) + 3 (- x^{2}) + 3 (4 x) + 3 \cdot - 3$

Step 1.8.9

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

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Step 1.8.9.1

Move $x$ .

$5 x^{3} + 28 x^{2} - 53 x - 28 - 4 x^{4} + x^{3} - 1 \cdot 4 (x \cdot x) - x \cdot - 3 + 3 (4 x^{3}) + 3 (- x^{2}) + 3 (4 x) + 3 \cdot - 3$

Step 1.8.9.2

Multiply $x$ by $x$ .

$5 x^{3} + 28 x^{2} - 53 x - 28 - 4 x^{4} + x^{3} - 1 \cdot 4 x^{2} - x \cdot - 3 + 3 (4 x^{3}) + 3 (- x^{2}) + 3 (4 x) + 3 \cdot - 3$

Step 1.8.10

Multiply $- 1$ by $4$ .

$5 x^{3} + 28 x^{2} - 53 x - 28 - 4 x^{4} + x^{3} - 4 x^{2} - x \cdot - 3 + 3 (4 x^{3}) + 3 (- x^{2}) + 3 (4 x) + 3 \cdot - 3$

Step 1.8.11

Multiply $- 3$ by $- 1$ .

$5 x^{3} + 28 x^{2} - 53 x - 28 - 4 x^{4} + x^{3} - 4 x^{2} + 3 x + 3 (4 x^{3}) + 3 (- x^{2}) + 3 (4 x) + 3 \cdot - 3$

Step 1.8.12

Multiply $4$ by $3$ .

$5 x^{3} + 28 x^{2} - 53 x - 28 - 4 x^{4} + x^{3} - 4 x^{2} + 3 x + 12 x^{3} + 3 (- x^{2}) + 3 (4 x) + 3 \cdot - 3$

Step 1.8.13

Multiply $- 1$ by $3$ .

$5 x^{3} + 28 x^{2} - 53 x - 28 - 4 x^{4} + x^{3} - 4 x^{2} + 3 x + 12 x^{3} - 3 x^{2} + 3 (4 x) + 3 \cdot - 3$

Step 1.8.14

Multiply $4$ by $3$ .

$5 x^{3} + 28 x^{2} - 53 x - 28 - 4 x^{4} + x^{3} - 4 x^{2} + 3 x + 12 x^{3} - 3 x^{2} + 12 x + 3 \cdot - 3$

Step 1.8.15

Multiply $3$ by $- 3$ .

$5 x^{3} + 28 x^{2} - 53 x - 28 - 4 x^{4} + x^{3} - 4 x^{2} + 3 x + 12 x^{3} - 3 x^{2} + 12 x - 9$

Step 1.9

Add $x^{3}$ and $12 x^{3}$ .

$5 x^{3} + 28 x^{2} - 53 x - 28 - 4 x^{4} + 13 x^{3} - 4 x^{2} + 3 x - 3 x^{2} + 12 x - 9$

Step 1.10

Subtract $3 x^{2}$ from $- 4 x^{2}$ .

$5 x^{3} + 28 x^{2} - 53 x - 28 - 4 x^{4} + 13 x^{3} - 7 x^{2} + 3 x + 12 x - 9$

Step 1.11

Add $3 x$ and $12 x$ .

$5 x^{3} + 28 x^{2} - 53 x - 28 - 4 x^{4} + 13 x^{3} - 7 x^{2} + 15 x - 9$

Step 2

Simplify by adding terms.

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Step 2.1

Add $5 x^{3}$ and $13 x^{3}$ .

$18 x^{3} + 28 x^{2} - 53 x - 28 - 4 x^{4} - 7 x^{2} + 15 x - 9$

Step 2.2

Subtract $7 x^{2}$ from $28 x^{2}$ .

$18 x^{3} + 21 x^{2} - 53 x - 28 - 4 x^{4} + 15 x - 9$

Step 2.3

Add $- 53 x$ and $15 x$ .

$18 x^{3} + 21 x^{2} - 38 x - 28 - 4 x^{4} - 9$

Step 2.4

Subtract $9$ from $- 28$ .

$18 x^{3} + 21 x^{2} - 38 x - 4 x^{4} - 37$