Algebra Examples

$f (x) = (x + 3) {(x - 2)}^{3}$

Step 1

Simplify and reorder the polynomial.

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Step 1.1

Use the Binomial Theorem.

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

Step 1.2

Simplify each term.

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Step 1.2.1

Multiply $- 2$ by $3$ .

$(x + 3) (x^{3} - 6 x^{2} + 3 x {(- 2)}^{2} + {(- 2)}^{3})$

Step 1.2.2

Raise $- 2$ to the power of $2$ .

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

Step 1.2.3

Multiply $4$ by $3$ .

$(x + 3) (x^{3} - 6 x^{2} + 12 x + {(- 2)}^{3})$

Step 1.2.4

Raise $- 2$ to the power of $3$ .

$(x + 3) (x^{3} - 6 x^{2} + 12 x - 8)$

Step 1.3

Expand $(x + 3) (x^{3} - 6 x^{2} + 12 x - 8)$ by multiplying each term in the first expression by each term in the second expression.

$x \cdot x^{3} + x (- 6 x^{2}) + x (12 x) + x \cdot - 8 + 3 x^{3} + 3 (- 6 x^{2}) + 3 (12 x) + 3 \cdot - 8$

Step 1.4

Simplify terms.

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Step 1.4.1

Simplify each term.

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Step 1.4.1.1

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

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Step 1.4.1.1.1

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

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Step 1.4.1.1.1.1

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

$x^{1} x^{3} + x (- 6 x^{2}) + x (12 x) + x \cdot - 8 + 3 x^{3} + 3 (- 6 x^{2}) + 3 (12 x) + 3 \cdot - 8$

Step 1.4.1.1.1.2

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

$x^{1 + 3} + x (- 6 x^{2}) + x (12 x) + x \cdot - 8 + 3 x^{3} + 3 (- 6 x^{2}) + 3 (12 x) + 3 \cdot - 8$

Step 1.4.1.1.2

Add $1$ and $3$ .

$x^{4} + x (- 6 x^{2}) + x (12 x) + x \cdot - 8 + 3 x^{3} + 3 (- 6 x^{2}) + 3 (12 x) + 3 \cdot - 8$

Step 1.4.1.2

Rewrite using the commutative property of multiplication.

$x^{4} - 6 x \cdot x^{2} + x (12 x) + x \cdot - 8 + 3 x^{3} + 3 (- 6 x^{2}) + 3 (12 x) + 3 \cdot - 8$

Step 1.4.1.3

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

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Step 1.4.1.3.1

Move $x^{2}$ .

$x^{4} - 6 (x^{2} x) + x (12 x) + x \cdot - 8 + 3 x^{3} + 3 (- 6 x^{2}) + 3 (12 x) + 3 \cdot - 8$

Step 1.4.1.3.2

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

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Step 1.4.1.3.2.1

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

$x^{4} - 6 (x^{2} x^{1}) + x (12 x) + x \cdot - 8 + 3 x^{3} + 3 (- 6 x^{2}) + 3 (12 x) + 3 \cdot - 8$

Step 1.4.1.3.2.2

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

$x^{4} - 6 x^{2 + 1} + x (12 x) + x \cdot - 8 + 3 x^{3} + 3 (- 6 x^{2}) + 3 (12 x) + 3 \cdot - 8$

Step 1.4.1.3.3

Add $2$ and $1$ .

$x^{4} - 6 x^{3} + x (12 x) + x \cdot - 8 + 3 x^{3} + 3 (- 6 x^{2}) + 3 (12 x) + 3 \cdot - 8$

Step 1.4.1.4

Rewrite using the commutative property of multiplication.

$x^{4} - 6 x^{3} + 12 x \cdot x + x \cdot - 8 + 3 x^{3} + 3 (- 6 x^{2}) + 3 (12 x) + 3 \cdot - 8$

Step 1.4.1.5

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

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Step 1.4.1.5.1

Move $x$ .

$x^{4} - 6 x^{3} + 12 (x \cdot x) + x \cdot - 8 + 3 x^{3} + 3 (- 6 x^{2}) + 3 (12 x) + 3 \cdot - 8$

Step 1.4.1.5.2

Multiply $x$ by $x$ .

$x^{4} - 6 x^{3} + 12 x^{2} + x \cdot - 8 + 3 x^{3} + 3 (- 6 x^{2}) + 3 (12 x) + 3 \cdot - 8$

Step 1.4.1.6

Move $- 8$ to the left of $x$ .

$x^{4} - 6 x^{3} + 12 x^{2} - 8 \cdot x + 3 x^{3} + 3 (- 6 x^{2}) + 3 (12 x) + 3 \cdot - 8$

Step 1.4.1.7

Multiply $- 6$ by $3$ .

$x^{4} - 6 x^{3} + 12 x^{2} - 8 x + 3 x^{3} - 18 x^{2} + 3 (12 x) + 3 \cdot - 8$

Step 1.4.1.8

Multiply $12$ by $3$ .

$x^{4} - 6 x^{3} + 12 x^{2} - 8 x + 3 x^{3} - 18 x^{2} + 36 x + 3 \cdot - 8$

Step 1.4.1.9

Multiply $3$ by $- 8$ .

$x^{4} - 6 x^{3} + 12 x^{2} - 8 x + 3 x^{3} - 18 x^{2} + 36 x - 24$

Step 1.4.2

Simplify by adding terms.

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Step 1.4.2.1

Add $- 6 x^{3}$ and $3 x^{3}$ .

$x^{4} - 3 x^{3} + 12 x^{2} - 8 x - 18 x^{2} + 36 x - 24$

Step 1.4.2.2

Subtract $18 x^{2}$ from $12 x^{2}$ .

$x^{4} - 3 x^{3} - 6 x^{2} - 8 x + 36 x - 24$

Step 1.4.2.3

Add $- 8 x$ and $36 x$ .

$x^{4} - 3 x^{3} - 6 x^{2} + 28 x - 24$

Step 2

The largest exponent is the degree of the polynomial.

$4$