Algebra Examples

$(5 x^{4} - 9 x^{3} + 7 x - 1) + (- 8 x^{4} + 4 x^{2} - 3 x + 2) - (- 4 x^{3} + 5 x - 1) (2 x - 7)$

Step 1

Remove parentheses.

$5 x^{4} - 9 x^{3} + 7 x - 1 - 8 x^{4} + 4 x^{2} - 3 x + 2 - (- 4 x^{3} + 5 x - 1) (2 x - 7)$

Step 2

Simplify each term.

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Step 2.1

Apply the distributive property.

$5 x^{4} - 9 x^{3} + 7 x - 1 - 8 x^{4} + 4 x^{2} - 3 x + 2 + (- (- 4 x^{3}) - (5 x) - - 1) (2 x - 7)$

Step 2.2

Simplify.

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Step 2.2.1

Multiply $- 4$ by $- 1$ .

$5 x^{4} - 9 x^{3} + 7 x - 1 - 8 x^{4} + 4 x^{2} - 3 x + 2 + (4 x^{3} - (5 x) - - 1) (2 x - 7)$

Step 2.2.2

Multiply $5$ by $- 1$ .

$5 x^{4} - 9 x^{3} + 7 x - 1 - 8 x^{4} + 4 x^{2} - 3 x + 2 + (4 x^{3} - 5 x - - 1) (2 x - 7)$

Step 2.2.3

Multiply $- 1$ by $- 1$ .

$5 x^{4} - 9 x^{3} + 7 x - 1 - 8 x^{4} + 4 x^{2} - 3 x + 2 + (4 x^{3} - 5 x + 1) (2 x - 7)$

Step 2.3

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

$5 x^{4} - 9 x^{3} + 7 x - 1 - 8 x^{4} + 4 x^{2} - 3 x + 2 + 4 x^{3} (2 x) + 4 x^{3} \cdot - 7 - 5 x (2 x) - 5 x \cdot - 7 + 1 (2 x) + 1 \cdot - 7$

Step 2.4

Simplify each term.

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Step 2.4.1

Rewrite using the commutative property of multiplication.

$5 x^{4} - 9 x^{3} + 7 x - 1 - 8 x^{4} + 4 x^{2} - 3 x + 2 + 4 \cdot 2 x^{3} x + 4 x^{3} \cdot - 7 - 5 x (2 x) - 5 x \cdot - 7 + 1 (2 x) + 1 \cdot - 7$

Step 2.4.2

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

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Step 2.4.2.1

Move $x$ .

$5 x^{4} - 9 x^{3} + 7 x - 1 - 8 x^{4} + 4 x^{2} - 3 x + 2 + 4 \cdot 2 (x \cdot x^{3}) + 4 x^{3} \cdot - 7 - 5 x (2 x) - 5 x \cdot - 7 + 1 (2 x) + 1 \cdot - 7$

Step 2.4.2.2

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

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Step 2.4.2.2.1

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

$5 x^{4} - 9 x^{3} + 7 x - 1 - 8 x^{4} + 4 x^{2} - 3 x + 2 + 4 \cdot 2 (x^{1} x^{3}) + 4 x^{3} \cdot - 7 - 5 x (2 x) - 5 x \cdot - 7 + 1 (2 x) + 1 \cdot - 7$

Step 2.4.2.2.2

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

$5 x^{4} - 9 x^{3} + 7 x - 1 - 8 x^{4} + 4 x^{2} - 3 x + 2 + 4 \cdot 2 x^{1 + 3} + 4 x^{3} \cdot - 7 - 5 x (2 x) - 5 x \cdot - 7 + 1 (2 x) + 1 \cdot - 7$

Step 2.4.2.3

Add $1$ and $3$ .

$5 x^{4} - 9 x^{3} + 7 x - 1 - 8 x^{4} + 4 x^{2} - 3 x + 2 + 4 \cdot 2 x^{4} + 4 x^{3} \cdot - 7 - 5 x (2 x) - 5 x \cdot - 7 + 1 (2 x) + 1 \cdot - 7$

Step 2.4.3

Multiply $4$ by $2$ .

$5 x^{4} - 9 x^{3} + 7 x - 1 - 8 x^{4} + 4 x^{2} - 3 x + 2 + 8 x^{4} + 4 x^{3} \cdot - 7 - 5 x (2 x) - 5 x \cdot - 7 + 1 (2 x) + 1 \cdot - 7$

Step 2.4.4

Multiply $- 7$ by $4$ .

$5 x^{4} - 9 x^{3} + 7 x - 1 - 8 x^{4} + 4 x^{2} - 3 x + 2 + 8 x^{4} - 28 x^{3} - 5 x (2 x) - 5 x \cdot - 7 + 1 (2 x) + 1 \cdot - 7$

Step 2.4.5

Rewrite using the commutative property of multiplication.

$5 x^{4} - 9 x^{3} + 7 x - 1 - 8 x^{4} + 4 x^{2} - 3 x + 2 + 8 x^{4} - 28 x^{3} - 5 \cdot 2 x \cdot x - 5 x \cdot - 7 + 1 (2 x) + 1 \cdot - 7$

Step 2.4.6

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

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Step 2.4.6.1

Move $x$ .

$5 x^{4} - 9 x^{3} + 7 x - 1 - 8 x^{4} + 4 x^{2} - 3 x + 2 + 8 x^{4} - 28 x^{3} - 5 \cdot 2 (x \cdot x) - 5 x \cdot - 7 + 1 (2 x) + 1 \cdot - 7$

Step 2.4.6.2

Multiply $x$ by $x$ .

$5 x^{4} - 9 x^{3} + 7 x - 1 - 8 x^{4} + 4 x^{2} - 3 x + 2 + 8 x^{4} - 28 x^{3} - 5 \cdot 2 x^{2} - 5 x \cdot - 7 + 1 (2 x) + 1 \cdot - 7$

Step 2.4.7

Multiply $- 5$ by $2$ .

$5 x^{4} - 9 x^{3} + 7 x - 1 - 8 x^{4} + 4 x^{2} - 3 x + 2 + 8 x^{4} - 28 x^{3} - 10 x^{2} - 5 x \cdot - 7 + 1 (2 x) + 1 \cdot - 7$

Step 2.4.8

Multiply $- 7$ by $- 5$ .

$5 x^{4} - 9 x^{3} + 7 x - 1 - 8 x^{4} + 4 x^{2} - 3 x + 2 + 8 x^{4} - 28 x^{3} - 10 x^{2} + 35 x + 1 (2 x) + 1 \cdot - 7$

Step 2.4.9

Multiply $2 x$ by $1$ .

$5 x^{4} - 9 x^{3} + 7 x - 1 - 8 x^{4} + 4 x^{2} - 3 x + 2 + 8 x^{4} - 28 x^{3} - 10 x^{2} + 35 x + 2 x + 1 \cdot - 7$

Step 2.4.10

Multiply $- 7$ by $1$ .

$5 x^{4} - 9 x^{3} + 7 x - 1 - 8 x^{4} + 4 x^{2} - 3 x + 2 + 8 x^{4} - 28 x^{3} - 10 x^{2} + 35 x + 2 x - 7$

Step 2.5

Add $35 x$ and $2 x$ .

$5 x^{4} - 9 x^{3} + 7 x - 1 - 8 x^{4} + 4 x^{2} - 3 x + 2 + 8 x^{4} - 28 x^{3} - 10 x^{2} + 37 x - 7$

Step 3

Simplify by adding terms.

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Step 3.1

Combine the opposite terms in $5 x^{4} - 9 x^{3} + 7 x - 1 - 8 x^{4} + 4 x^{2} - 3 x + 2 + 8 x^{4} - 28 x^{3} - 10 x^{2} + 37 x - 7$ .

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Step 3.1.1

Add $- 8 x^{4}$ and $8 x^{4}$ .

$5 x^{4} - 9 x^{3} + 7 x - 1 + 4 x^{2} - 3 x + 2 + 0 - 28 x^{3} - 10 x^{2} + 37 x - 7$

Step 3.1.2

Add $5 x^{4} - 9 x^{3} + 7 x - 1 + 4 x^{2} - 3 x + 2$ and $0$ .

$5 x^{4} - 9 x^{3} + 7 x - 1 + 4 x^{2} - 3 x + 2 - 28 x^{3} - 10 x^{2} + 37 x - 7$

Step 3.2

Subtract $28 x^{3}$ from $- 9 x^{3}$ .

$5 x^{4} - 37 x^{3} + 7 x - 1 + 4 x^{2} - 3 x + 2 - 10 x^{2} + 37 x - 7$

Step 3.3

Subtract $3 x$ from $7 x$ .

$5 x^{4} - 37 x^{3} + 4 x - 1 + 4 x^{2} + 2 - 10 x^{2} + 37 x - 7$

Step 3.4

Add $4 x$ and $37 x$ .

$5 x^{4} - 37 x^{3} + 41 x - 1 + 4 x^{2} + 2 - 10 x^{2} - 7$

Step 3.5

Add $- 1$ and $2$ .

$5 x^{4} - 37 x^{3} + 41 x + 4 x^{2} + 1 - 10 x^{2} - 7$

Step 3.6

Subtract $10 x^{2}$ from $4 x^{2}$ .

$5 x^{4} - 37 x^{3} + 41 x - 6 x^{2} + 1 - 7$

Step 3.7

Simplify the expression.

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Step 3.7.1

Subtract $7$ from $1$ .

$5 x^{4} - 37 x^{3} + 41 x - 6 x^{2} - 6$

Step 3.7.2

Move $41 x$ .

$5 x^{4} - 37 x^{3} - 6 x^{2} + 41 x - 6$