Algebra Examples

$y = (x + 4) (x + 1) {(x - 1)}^{2}$

Step 1

To write a polynomial in standard form, simplify and then arrange the terms in descending order.

$y = a x^{2} + b x + c$

Step 2

Expand $(x + 4) (x + 1)$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

$y = (x (x + 1) + 4 (x + 1)) {(x - 1)}^{2}$

Step 2.2

Apply the distributive property.

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

Step 2.3

Apply the distributive property.

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

Step 3

Simplify and combine like terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

Multiply $x$ by $x$ .

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

Step 3.1.2

Multiply $x$ by $1$ .

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

Step 3.1.3

Multiply $4$ by $1$ .

$y = (x^{2} + x + 4 x + 4) {(x - 1)}^{2}$

Step 3.2

Add $x$ and $4 x$ .

$y = (x^{2} + 5 x + 4) {(x - 1)}^{2}$

Step 4

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

$y = (x^{2} + 5 x + 4) ((x - 1) (x - 1))$

Step 5

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

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Step 5.1

Apply the distributive property.

$y = (x^{2} + 5 x + 4) (x (x - 1) - 1 (x - 1))$

Step 5.2

Apply the distributive property.

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

Step 5.3

Apply the distributive property.

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

Step 6

Simplify and combine like terms.

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Step 6.1

Simplify each term.

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Step 6.1.1

Multiply $x$ by $x$ .

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

Step 6.1.2

Move $- 1$ to the left of $x$ .

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

Step 6.1.3

Rewrite $- 1 x$ as $- x$ .

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

Step 6.1.4

Rewrite $- 1 x$ as $- x$ .

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

Step 6.1.5

Multiply $- 1$ by $- 1$ .

$y = (x^{2} + 5 x + 4) (x^{2} - x - x + 1)$

Step 6.2

Subtract $x$ from $- x$ .

$y = (x^{2} + 5 x + 4) (x^{2} - 2 x + 1)$

Step 7

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

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

Step 8

Simplify terms.

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Step 8.1

Simplify each term.

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Step 8.1.1

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

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Step 8.1.1.1

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

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

Step 8.1.1.2

Add $2$ and $2$ .

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

Step 8.1.2

Rewrite using the commutative property of multiplication.

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

Step 8.1.3

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

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Step 8.1.3.1

Move $x$ .

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

Step 8.1.3.2

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

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Step 8.1.3.2.1

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

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

Step 8.1.3.2.2

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

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

Step 8.1.3.3

Add $1$ and $2$ .

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

Step 8.1.4

Multiply $x^{2}$ by $1$ .

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

Step 8.1.5

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

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Step 8.1.5.1

Move $x^{2}$ .

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

Step 8.1.5.2

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

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Step 8.1.5.2.1

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

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

Step 8.1.5.2.2

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

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

Step 8.1.5.3

Add $2$ and $1$ .

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

Step 8.1.6

Rewrite using the commutative property of multiplication.

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

Step 8.1.7

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

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Step 8.1.7.1

Move $x$ .

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

Step 8.1.7.2

Multiply $x$ by $x$ .

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

Step 8.1.8

Multiply $5$ by $- 2$ .

$y = x^{4} - 2 x^{3} + x^{2} + 5 x^{3} - 10 x^{2} + 5 x \cdot 1 + 4 x^{2} + 4 (- 2 x) + 4 \cdot 1$

Step 8.1.9

Multiply $5$ by $1$ .

$y = x^{4} - 2 x^{3} + x^{2} + 5 x^{3} - 10 x^{2} + 5 x + 4 x^{2} + 4 (- 2 x) + 4 \cdot 1$

Step 8.1.10

Multiply $- 2$ by $4$ .

$y = x^{4} - 2 x^{3} + x^{2} + 5 x^{3} - 10 x^{2} + 5 x + 4 x^{2} - 8 x + 4 \cdot 1$

Step 8.1.11

Multiply $4$ by $1$ .

$y = x^{4} - 2 x^{3} + x^{2} + 5 x^{3} - 10 x^{2} + 5 x + 4 x^{2} - 8 x + 4$

Step 8.2

Simplify by adding terms.

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Step 8.2.1

Add $- 2 x^{3}$ and $5 x^{3}$ .

$y = x^{4} + 3 x^{3} + x^{2} - 10 x^{2} + 5 x + 4 x^{2} - 8 x + 4$

Step 8.2.2

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

$y = x^{4} + 3 x^{3} - 9 x^{2} + 5 x + 4 x^{2} - 8 x + 4$

Step 8.2.3

Add $- 9 x^{2}$ and $4 x^{2}$ .

$y = x^{4} + 3 x^{3} - 5 x^{2} + 5 x - 8 x + 4$

Step 8.2.4

Subtract $8 x$ from $5 x$ .

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

Step 9