Algebra Examples

$y = - (x + 3) {(x + 1)}^{2} (2 x - 5)$

Step 1

Simplify the polynomial, then reorder it left to right starting with the highest degree term.

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Step 1.1

Simplify by multiplying through.

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Step 1.1.1

Apply the distributive property.

$y = (- x - 1 \cdot 3) {(x + 1)}^{2} (2 x - 5)$

Step 1.1.2

Simplify the expression.

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Step 1.1.2.1

Multiply $- 1$ by $3$ .

$y = (- x - 3) {(x + 1)}^{2} (2 x - 5)$

Step 1.1.2.2

Rewrite ${(x + 1)}^{2}$ as $(x + 1) (x + 1)$ .

$y = (- x - 3) ((x + 1) (x + 1)) (2 x - 5)$

Step 1.2

Expand $(x + 1) (x + 1)$ using the FOIL Method.

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Step 1.2.1

Apply the distributive property.

$y = (- x - 3) (x (x + 1) + 1 (x + 1)) (2 x - 5)$

Step 1.2.2

Apply the distributive property.

$y = (- x - 3) (x \cdot x + x \cdot 1 + 1 (x + 1)) (2 x - 5)$

Step 1.2.3

Apply the distributive property.

$y = (- x - 3) (x \cdot x + x \cdot 1 + 1 x + 1 \cdot 1) (2 x - 5)$

Step 1.3

Simplify and combine like terms.

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

Multiply $x$ by $x$ .

$y = (- x - 3) (x^{2} + x \cdot 1 + 1 x + 1 \cdot 1) (2 x - 5)$

Step 1.3.1.2

Multiply $x$ by $1$ .

$y = (- x - 3) (x^{2} + x + 1 x + 1 \cdot 1) (2 x - 5)$

Step 1.3.1.3

Multiply $x$ by $1$ .

$y = (- x - 3) (x^{2} + x + x + 1 \cdot 1) (2 x - 5)$

Step 1.3.1.4

Multiply $1$ by $1$ .

$y = (- x - 3) (x^{2} + x + x + 1) (2 x - 5)$

Step 1.3.2

Add $x$ and $x$ .

$y = (- x - 3) (x^{2} + 2 x + 1) (2 x - 5)$

Step 1.4

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

$y = (- x \cdot x^{2} - x (2 x) - x \cdot 1 - 3 x^{2} - 3 (2 x) - 3 \cdot 1) (2 x - 5)$

Step 1.5

Simplify terms.

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Step 1.5.1

Simplify each term.

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Step 1.5.1.1

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

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Step 1.5.1.1.1

Move $x^{2}$ .

$y = (- (x^{2} x) - x (2 x) - x \cdot 1 - 3 x^{2} - 3 (2 x) - 3 \cdot 1) (2 x - 5)$

Step 1.5.1.1.2

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

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Step 1.5.1.1.2.1

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

$y = (- (x^{2} x) - x (2 x) - x \cdot 1 - 3 x^{2} - 3 (2 x) - 3 \cdot 1) (2 x - 5)$

Step 1.5.1.1.2.2

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

$y = (- x^{2 + 1} - x (2 x) - x \cdot 1 - 3 x^{2} - 3 (2 x) - 3 \cdot 1) (2 x - 5)$

Step 1.5.1.1.3

Add $2$ and $1$ .

$y = (- x^{3} - x (2 x) - x \cdot 1 - 3 x^{2} - 3 (2 x) - 3 \cdot 1) (2 x - 5)$

Step 1.5.1.2

Rewrite using the commutative property of multiplication.

$y = (- x^{3} - 1 \cdot (2 x \cdot x) - x \cdot 1 - 3 x^{2} - 3 (2 x) - 3 \cdot 1) (2 x - 5)$

Step 1.5.1.3

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

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Step 1.5.1.3.1

Move $x$ .

$y = (- x^{3} - 1 \cdot (2 (x \cdot x)) - x \cdot 1 - 3 x^{2} - 3 (2 x) - 3 \cdot 1) (2 x - 5)$

Step 1.5.1.3.2

Multiply $x$ by $x$ .

$y = (- x^{3} - 1 \cdot (2 x^{2}) - x \cdot 1 - 3 x^{2} - 3 (2 x) - 3 \cdot 1) (2 x - 5)$

Step 1.5.1.4

Multiply $- 1$ by $2$ .

$y = (- x^{3} - 2 x^{2} - x \cdot 1 - 3 x^{2} - 3 (2 x) - 3 \cdot 1) (2 x - 5)$

Step 1.5.1.5

Multiply $- 1$ by $1$ .

$y = (- x^{3} - 2 x^{2} - x - 3 x^{2} - 3 (2 x) - 3 \cdot 1) (2 x - 5)$

Step 1.5.1.6

Multiply $2$ by $- 3$ .

$y = (- x^{3} - 2 x^{2} - x - 3 x^{2} - 6 x - 3 \cdot 1) (2 x - 5)$

Step 1.5.1.7

Multiply $- 3$ by $1$ .

$y = (- x^{3} - 2 x^{2} - x - 3 x^{2} - 6 x - 3) (2 x - 5)$

Step 1.5.2

Simplify by adding terms.

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Step 1.5.2.1

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

$y = (- x^{3} - 5 x^{2} - x - 6 x - 3) (2 x - 5)$

Step 1.5.2.2

Subtract $6 x$ from $- x$ .

$y = (- x^{3} - 5 x^{2} - 7 x - 3) (2 x - 5)$

Step 1.6

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

$y = - x^{3} (2 x) - x^{3} \cdot - 5 - 5 x^{2} (2 x) - 5 x^{2} \cdot - 5 - 7 x (2 x) - 7 x \cdot - 5 - 3 (2 x) - 3 \cdot - 5$

Step 1.7

Simplify terms.

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Step 1.7.1

Simplify each term.

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Step 1.7.1.1

Rewrite using the commutative property of multiplication.

$y = - 1 \cdot (2 x^{3} x) - x^{3} \cdot - 5 - 5 x^{2} (2 x) - 5 x^{2} \cdot - 5 - 7 x (2 x) - 7 x \cdot - 5 - 3 (2 x) - 3 \cdot - 5$

Step 1.7.1.2

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

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Step 1.7.1.2.1

Move $x$ .

$y = - 1 \cdot (2 (x \cdot x^{3})) - x^{3} \cdot - 5 - 5 x^{2} (2 x) - 5 x^{2} \cdot - 5 - 7 x (2 x) - 7 x \cdot - 5 - 3 (2 x) - 3 \cdot - 5$

Step 1.7.1.2.2

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

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Step 1.7.1.2.2.1

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

$y = - 1 \cdot (2 (x \cdot x^{3})) - x^{3} \cdot - 5 - 5 x^{2} (2 x) - 5 x^{2} \cdot - 5 - 7 x (2 x) - 7 x \cdot - 5 - 3 (2 x) - 3 \cdot - 5$

Step 1.7.1.2.2.2

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

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

Step 1.7.1.2.3

Add $1$ and $3$ .

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

Step 1.7.1.3

Multiply $- 1$ by $2$ .

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

Step 1.7.1.4

Multiply $- 5$ by $- 1$ .

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

Step 1.7.1.5

Rewrite using the commutative property of multiplication.

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

Step 1.7.1.6

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

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Step 1.7.1.6.1

Move $x$ .

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

Step 1.7.1.6.2

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

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Step 1.7.1.6.2.1

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

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

Step 1.7.1.6.2.2

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

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

Step 1.7.1.6.3

Add $1$ and $2$ .

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

Step 1.7.1.7

Multiply $- 5$ by $2$ .

$y = - 2 x^{4} + 5 x^{3} - 10 x^{3} - 5 x^{2} \cdot - 5 - 7 x (2 x) - 7 x \cdot - 5 - 3 (2 x) - 3 \cdot - 5$

Step 1.7.1.8

Multiply $- 5$ by $- 5$ .

$y = - 2 x^{4} + 5 x^{3} - 10 x^{3} + 25 x^{2} - 7 x (2 x) - 7 x \cdot - 5 - 3 (2 x) - 3 \cdot - 5$

Step 1.7.1.9

Rewrite using the commutative property of multiplication.

$y = - 2 x^{4} + 5 x^{3} - 10 x^{3} + 25 x^{2} - 7 \cdot (2 x \cdot x) - 7 x \cdot - 5 - 3 (2 x) - 3 \cdot - 5$

Step 1.7.1.10

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

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Step 1.7.1.10.1

Move $x$ .

$y = - 2 x^{4} + 5 x^{3} - 10 x^{3} + 25 x^{2} - 7 \cdot (2 (x \cdot x)) - 7 x \cdot - 5 - 3 (2 x) - 3 \cdot - 5$

Step 1.7.1.10.2

Multiply $x$ by $x$ .

$y = - 2 x^{4} + 5 x^{3} - 10 x^{3} + 25 x^{2} - 7 \cdot (2 x^{2}) - 7 x \cdot - 5 - 3 (2 x) - 3 \cdot - 5$

Step 1.7.1.11

Multiply $- 7$ by $2$ .

$y = - 2 x^{4} + 5 x^{3} - 10 x^{3} + 25 x^{2} - 14 x^{2} - 7 x \cdot - 5 - 3 (2 x) - 3 \cdot - 5$

Step 1.7.1.12

Multiply $- 5$ by $- 7$ .

$y = - 2 x^{4} + 5 x^{3} - 10 x^{3} + 25 x^{2} - 14 x^{2} + 35 x - 3 (2 x) - 3 \cdot - 5$

Step 1.7.1.13

Multiply $2$ by $- 3$ .

$y = - 2 x^{4} + 5 x^{3} - 10 x^{3} + 25 x^{2} - 14 x^{2} + 35 x - 6 x - 3 \cdot - 5$

Step 1.7.1.14

Multiply $- 3$ by $- 5$ .

$y = - 2 x^{4} + 5 x^{3} - 10 x^{3} + 25 x^{2} - 14 x^{2} + 35 x - 6 x + 15$

Step 1.7.2

Simplify by adding terms.

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Step 1.7.2.1

Subtract $10 x^{3}$ from $5 x^{3}$ .

$y = - 2 x^{4} - 5 x^{3} + 25 x^{2} - 14 x^{2} + 35 x - 6 x + 15$

Step 1.7.2.2

Subtract $14 x^{2}$ from $25 x^{2}$ .

$y = - 2 x^{4} - 5 x^{3} + 11 x^{2} + 35 x - 6 x + 15$

Step 1.7.2.3

Subtract $6 x$ from $35 x$ .

$y = - 2 x^{4} - 5 x^{3} + 11 x^{2} + 29 x + 15$

Step 2

The leading term in a polynomial is the term with the highest degree.

$- 2 x^{4}$