Precalculus Examples

$(x - 1) {(x - 2)}^{2} {(x - p)}^{3}$

Step 1

Simplify and reorder the polynomial.

Tap for more steps...

Step 1.1

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

$(x - 1) ((x - 2) (x - 2)) {(x - p)}^{3}$

Step 1.2

Expand $(x - 2) (x - 2)$ using the FOIL Method.

Tap for more steps...

Step 1.2.1

Apply the distributive property.

$(x - 1) (x (x - 2) - 2 (x - 2)) {(x - p)}^{3}$

Step 1.2.2

Apply the distributive property.

$(x - 1) (x \cdot x + x \cdot - 2 - 2 (x - 2)) {(x - p)}^{3}$

Step 1.2.3

Apply the distributive property.

$(x - 1) (x \cdot x + x \cdot - 2 - 2 x - 2 \cdot - 2) {(x - p)}^{3}$

Step 1.3

Simplify and combine like terms.

Tap for more steps...

Step 1.3.1

Simplify each term.

Tap for more steps...

Step 1.3.1.1

Multiply $x$ by $x$ .

$(x - 1) (x^{2} + x \cdot - 2 - 2 x - 2 \cdot - 2) {(x - p)}^{3}$

Step 1.3.1.2

Move $- 2$ to the left of $x$ .

$(x - 1) (x^{2} - 2 \cdot x - 2 x - 2 \cdot - 2) {(x - p)}^{3}$

Step 1.3.1.3

Multiply $- 2$ by $- 2$ .

$(x - 1) (x^{2} - 2 x - 2 x + 4) {(x - p)}^{3}$

Step 1.3.2

Subtract $2 x$ from $- 2 x$ .

$(x - 1) (x^{2} - 4 x + 4) {(x - p)}^{3}$

Step 1.4

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

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

Step 1.5

Simplify terms.

Tap for more steps...

Step 1.5.1

Simplify each term.

Tap for more steps...

Step 1.5.1.1

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

Tap for more steps...

Step 1.5.1.1.1

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

Tap for more steps...

Step 1.5.1.1.1.1

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

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

Step 1.5.1.1.1.2

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

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

Step 1.5.1.1.2

Add $1$ and $2$ .

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

Step 1.5.1.2

Rewrite using the commutative property of multiplication.

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

Step 1.5.1.3

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

Tap for more steps...

Step 1.5.1.3.1

Move $x$ .

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

Step 1.5.1.3.2

Multiply $x$ by $x$ .

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

Step 1.5.1.4

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

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

Step 1.5.1.5

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

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

Step 1.5.1.6

Multiply $- 4$ by $- 1$ .

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

Step 1.5.1.7

Multiply $- 1$ by $4$ .

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

Step 1.5.2

Simplify by adding terms.

Tap for more steps...

Step 1.5.2.1

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

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

Step 1.5.2.2

Add $4 x$ and $4 x$ .

$(x^{3} - 5 x^{2} + 8 x - 4) {(x - p)}^{3}$

Step 1.6

Use the Binomial Theorem.

$(x^{3} - 5 x^{2} + 8 x - 4) (x^{3} + 3 x^{2} (- p) + 3 x {(- p)}^{2} + {(- p)}^{3})$

Step 1.7

Simplify each term.

Tap for more steps...

Step 1.7.1

Rewrite using the commutative property of multiplication.

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

Step 1.7.2

Multiply $3$ by $- 1$ .

$(x^{3} - 5 x^{2} + 8 x - 4) (x^{3} - 3 x^{2} p + 3 x {(- p)}^{2} + {(- p)}^{3})$

Step 1.7.3

Apply the product rule to $- p$ .

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

Step 1.7.4

Rewrite using the commutative property of multiplication.

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

Step 1.7.5

Raise $- 1$ to the power of $2$ .

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

Step 1.7.6

Multiply $3$ by $1$ .

$(x^{3} - 5 x^{2} + 8 x - 4) (x^{3} - 3 x^{2} p + 3 x p^{2} + {(- p)}^{3})$

Step 1.7.7

Apply the product rule to $- p$ .

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

Step 1.7.8

Raise $- 1$ to the power of $3$ .

$(x^{3} - 5 x^{2} + 8 x - 4) (x^{3} - 3 x^{2} p + 3 x p^{2} - p^{3})$

Step 1.8

Expand $(x^{3} - 5 x^{2} + 8 x - 4) (x^{3} - 3 x^{2} p + 3 x p^{2} - p^{3})$ by multiplying each term in the first expression by each term in the second expression.

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

Step 1.9

Simplify each term.

Tap for more steps...

Step 1.9.1

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

Tap for more steps...

Step 1.9.1.1

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

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

Step 1.9.1.2

Add $3$ and $3$ .

$x^{6} + x^{3} (- 3 x^{2} p) + x^{3} (3 x p^{2}) + x^{3} (- p^{3}) - 5 x^{2} x^{3} - 5 x^{2} (- 3 x^{2} p) - 5 x^{2} (3 x p^{2}) - 5 x^{2} (- p^{3}) + 8 x \cdot x^{3} + 8 x (- 3 x^{2} p) + 8 x (3 x p^{2}) + 8 x (- p^{3}) - 4 x^{3} - 4 (- 3 x^{2} p) - 4 (3 x p^{2}) - 4 (- p^{3})$

Step 1.9.2

Rewrite using the commutative property of multiplication.

$x^{6} - 3 x^{3} (x^{2} p) + x^{3} (3 x p^{2}) + x^{3} (- p^{3}) - 5 x^{2} x^{3} - 5 x^{2} (- 3 x^{2} p) - 5 x^{2} (3 x p^{2}) - 5 x^{2} (- p^{3}) + 8 x \cdot x^{3} + 8 x (- 3 x^{2} p) + 8 x (3 x p^{2}) + 8 x (- p^{3}) - 4 x^{3} - 4 (- 3 x^{2} p) - 4 (3 x p^{2}) - 4 (- p^{3})$

Step 1.9.3

Multiply $x^{3}$ by $x^{2}$ by adding the exponents.

Tap for more steps...

Step 1.9.3.1

Move $x^{2}$ .

$x^{6} - 3 (x^{2} x^{3}) p + x^{3} (3 x p^{2}) + x^{3} (- p^{3}) - 5 x^{2} x^{3} - 5 x^{2} (- 3 x^{2} p) - 5 x^{2} (3 x p^{2}) - 5 x^{2} (- p^{3}) + 8 x \cdot x^{3} + 8 x (- 3 x^{2} p) + 8 x (3 x p^{2}) + 8 x (- p^{3}) - 4 x^{3} - 4 (- 3 x^{2} p) - 4 (3 x p^{2}) - 4 (- p^{3})$

Step 1.9.3.2

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

$x^{6} - 3 x^{2 + 3} p + x^{3} (3 x p^{2}) + x^{3} (- p^{3}) - 5 x^{2} x^{3} - 5 x^{2} (- 3 x^{2} p) - 5 x^{2} (3 x p^{2}) - 5 x^{2} (- p^{3}) + 8 x \cdot x^{3} + 8 x (- 3 x^{2} p) + 8 x (3 x p^{2}) + 8 x (- p^{3}) - 4 x^{3} - 4 (- 3 x^{2} p) - 4 (3 x p^{2}) - 4 (- p^{3})$

Step 1.9.3.3

Add $2$ and $3$ .

$x^{6} - 3 x^{5} p + x^{3} (3 x p^{2}) + x^{3} (- p^{3}) - 5 x^{2} x^{3} - 5 x^{2} (- 3 x^{2} p) - 5 x^{2} (3 x p^{2}) - 5 x^{2} (- p^{3}) + 8 x \cdot x^{3} + 8 x (- 3 x^{2} p) + 8 x (3 x p^{2}) + 8 x (- p^{3}) - 4 x^{3} - 4 (- 3 x^{2} p) - 4 (3 x p^{2}) - 4 (- p^{3})$

Step 1.9.4

Rewrite using the commutative property of multiplication.

$x^{6} - 3 x^{5} p + 3 x^{3} (x p^{2}) + x^{3} (- p^{3}) - 5 x^{2} x^{3} - 5 x^{2} (- 3 x^{2} p) - 5 x^{2} (3 x p^{2}) - 5 x^{2} (- p^{3}) + 8 x \cdot x^{3} + 8 x (- 3 x^{2} p) + 8 x (3 x p^{2}) + 8 x (- p^{3}) - 4 x^{3} - 4 (- 3 x^{2} p) - 4 (3 x p^{2}) - 4 (- p^{3})$

Step 1.9.5

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

Tap for more steps...

Step 1.9.5.1

Move $x$ .

$x^{6} - 3 x^{5} p + 3 (x \cdot x^{3}) p^{2} + x^{3} (- p^{3}) - 5 x^{2} x^{3} - 5 x^{2} (- 3 x^{2} p) - 5 x^{2} (3 x p^{2}) - 5 x^{2} (- p^{3}) + 8 x \cdot x^{3} + 8 x (- 3 x^{2} p) + 8 x (3 x p^{2}) + 8 x (- p^{3}) - 4 x^{3} - 4 (- 3 x^{2} p) - 4 (3 x p^{2}) - 4 (- p^{3})$

Step 1.9.5.2

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

Tap for more steps...

Step 1.9.5.2.1

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

$x^{6} - 3 x^{5} p + 3 (x^{1} x^{3}) p^{2} + x^{3} (- p^{3}) - 5 x^{2} x^{3} - 5 x^{2} (- 3 x^{2} p) - 5 x^{2} (3 x p^{2}) - 5 x^{2} (- p^{3}) + 8 x \cdot x^{3} + 8 x (- 3 x^{2} p) + 8 x (3 x p^{2}) + 8 x (- p^{3}) - 4 x^{3} - 4 (- 3 x^{2} p) - 4 (3 x p^{2}) - 4 (- p^{3})$

Step 1.9.5.2.2

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

$x^{6} - 3 x^{5} p + 3 x^{1 + 3} p^{2} + x^{3} (- p^{3}) - 5 x^{2} x^{3} - 5 x^{2} (- 3 x^{2} p) - 5 x^{2} (3 x p^{2}) - 5 x^{2} (- p^{3}) + 8 x \cdot x^{3} + 8 x (- 3 x^{2} p) + 8 x (3 x p^{2}) + 8 x (- p^{3}) - 4 x^{3} - 4 (- 3 x^{2} p) - 4 (3 x p^{2}) - 4 (- p^{3})$

Step 1.9.5.3

Add $1$ and $3$ .

$x^{6} - 3 x^{5} p + 3 x^{4} p^{2} + x^{3} (- p^{3}) - 5 x^{2} x^{3} - 5 x^{2} (- 3 x^{2} p) - 5 x^{2} (3 x p^{2}) - 5 x^{2} (- p^{3}) + 8 x \cdot x^{3} + 8 x (- 3 x^{2} p) + 8 x (3 x p^{2}) + 8 x (- p^{3}) - 4 x^{3} - 4 (- 3 x^{2} p) - 4 (3 x p^{2}) - 4 (- p^{3})$

Step 1.9.6

Rewrite using the commutative property of multiplication.

$x^{6} - 3 x^{5} p + 3 x^{4} p^{2} - x^{3} p^{3} - 5 x^{2} x^{3} - 5 x^{2} (- 3 x^{2} p) - 5 x^{2} (3 x p^{2}) - 5 x^{2} (- p^{3}) + 8 x \cdot x^{3} + 8 x (- 3 x^{2} p) + 8 x (3 x p^{2}) + 8 x (- p^{3}) - 4 x^{3} - 4 (- 3 x^{2} p) - 4 (3 x p^{2}) - 4 (- p^{3})$

Step 1.9.7

Multiply $x^{2}$ by $x^{3}$ by adding the exponents.

Tap for more steps...

Step 1.9.7.1

Move $x^{3}$ .

$x^{6} - 3 x^{5} p + 3 x^{4} p^{2} - x^{3} p^{3} - 5 (x^{3} x^{2}) - 5 x^{2} (- 3 x^{2} p) - 5 x^{2} (3 x p^{2}) - 5 x^{2} (- p^{3}) + 8 x \cdot x^{3} + 8 x (- 3 x^{2} p) + 8 x (3 x p^{2}) + 8 x (- p^{3}) - 4 x^{3} - 4 (- 3 x^{2} p) - 4 (3 x p^{2}) - 4 (- p^{3})$

Step 1.9.7.2

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

$x^{6} - 3 x^{5} p + 3 x^{4} p^{2} - x^{3} p^{3} - 5 x^{3 + 2} - 5 x^{2} (- 3 x^{2} p) - 5 x^{2} (3 x p^{2}) - 5 x^{2} (- p^{3}) + 8 x \cdot x^{3} + 8 x (- 3 x^{2} p) + 8 x (3 x p^{2}) + 8 x (- p^{3}) - 4 x^{3} - 4 (- 3 x^{2} p) - 4 (3 x p^{2}) - 4 (- p^{3})$

Step 1.9.7.3

Add $3$ and $2$ .

$x^{6} - 3 x^{5} p + 3 x^{4} p^{2} - x^{3} p^{3} - 5 x^{5} - 5 x^{2} (- 3 x^{2} p) - 5 x^{2} (3 x p^{2}) - 5 x^{2} (- p^{3}) + 8 x \cdot x^{3} + 8 x (- 3 x^{2} p) + 8 x (3 x p^{2}) + 8 x (- p^{3}) - 4 x^{3} - 4 (- 3 x^{2} p) - 4 (3 x p^{2}) - 4 (- p^{3})$

Step 1.9.8

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

Tap for more steps...

Step 1.9.8.1

Move $x^{2}$ .

$x^{6} - 3 x^{5} p + 3 x^{4} p^{2} - x^{3} p^{3} - 5 x^{5} - 5 (x^{2} x^{2}) (- 3 p) - 5 x^{2} (3 x p^{2}) - 5 x^{2} (- p^{3}) + 8 x \cdot x^{3} + 8 x (- 3 x^{2} p) + 8 x (3 x p^{2}) + 8 x (- p^{3}) - 4 x^{3} - 4 (- 3 x^{2} p) - 4 (3 x p^{2}) - 4 (- p^{3})$

Step 1.9.8.2

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

$x^{6} - 3 x^{5} p + 3 x^{4} p^{2} - x^{3} p^{3} - 5 x^{5} - 5 x^{2 + 2} (- 3 p) - 5 x^{2} (3 x p^{2}) - 5 x^{2} (- p^{3}) + 8 x \cdot x^{3} + 8 x (- 3 x^{2} p) + 8 x (3 x p^{2}) + 8 x (- p^{3}) - 4 x^{3} - 4 (- 3 x^{2} p) - 4 (3 x p^{2}) - 4 (- p^{3})$

Step 1.9.8.3

Add $2$ and $2$ .

$x^{6} - 3 x^{5} p + 3 x^{4} p^{2} - x^{3} p^{3} - 5 x^{5} - 5 x^{4} (- 3 p) - 5 x^{2} (3 x p^{2}) - 5 x^{2} (- p^{3}) + 8 x \cdot x^{3} + 8 x (- 3 x^{2} p) + 8 x (3 x p^{2}) + 8 x (- p^{3}) - 4 x^{3} - 4 (- 3 x^{2} p) - 4 (3 x p^{2}) - 4 (- p^{3})$

Step 1.9.9

Rewrite using the commutative property of multiplication.

$x^{6} - 3 x^{5} p + 3 x^{4} p^{2} - x^{3} p^{3} - 5 x^{5} - 5 \cdot - 3 x^{4} p - 5 x^{2} (3 x p^{2}) - 5 x^{2} (- p^{3}) + 8 x \cdot x^{3} + 8 x (- 3 x^{2} p) + 8 x (3 x p^{2}) + 8 x (- p^{3}) - 4 x^{3} - 4 (- 3 x^{2} p) - 4 (3 x p^{2}) - 4 (- p^{3})$

Step 1.9.10

Multiply $- 5$ by $- 3$ .

$x^{6} - 3 x^{5} p + 3 x^{4} p^{2} - x^{3} p^{3} - 5 x^{5} + 15 x^{4} p - 5 x^{2} (3 x p^{2}) - 5 x^{2} (- p^{3}) + 8 x \cdot x^{3} + 8 x (- 3 x^{2} p) + 8 x (3 x p^{2}) + 8 x (- p^{3}) - 4 x^{3} - 4 (- 3 x^{2} p) - 4 (3 x p^{2}) - 4 (- p^{3})$

Step 1.9.11

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

Tap for more steps...

Step 1.9.11.1

Move $x$ .

$x^{6} - 3 x^{5} p + 3 x^{4} p^{2} - x^{3} p^{3} - 5 x^{5} + 15 x^{4} p - 5 (x \cdot x^{2}) (3 p^{2}) - 5 x^{2} (- p^{3}) + 8 x \cdot x^{3} + 8 x (- 3 x^{2} p) + 8 x (3 x p^{2}) + 8 x (- p^{3}) - 4 x^{3} - 4 (- 3 x^{2} p) - 4 (3 x p^{2}) - 4 (- p^{3})$

Step 1.9.11.2

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

Tap for more steps...

Step 1.9.11.2.1

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

$x^{6} - 3 x^{5} p + 3 x^{4} p^{2} - x^{3} p^{3} - 5 x^{5} + 15 x^{4} p - 5 (x^{1} x^{2}) (3 p^{2}) - 5 x^{2} (- p^{3}) + 8 x \cdot x^{3} + 8 x (- 3 x^{2} p) + 8 x (3 x p^{2}) + 8 x (- p^{3}) - 4 x^{3} - 4 (- 3 x^{2} p) - 4 (3 x p^{2}) - 4 (- p^{3})$

Step 1.9.11.2.2

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

$x^{6} - 3 x^{5} p + 3 x^{4} p^{2} - x^{3} p^{3} - 5 x^{5} + 15 x^{4} p - 5 x^{1 + 2} (3 p^{2}) - 5 x^{2} (- p^{3}) + 8 x \cdot x^{3} + 8 x (- 3 x^{2} p) + 8 x (3 x p^{2}) + 8 x (- p^{3}) - 4 x^{3} - 4 (- 3 x^{2} p) - 4 (3 x p^{2}) - 4 (- p^{3})$

Step 1.9.11.3

Add $1$ and $2$ .

$x^{6} - 3 x^{5} p + 3 x^{4} p^{2} - x^{3} p^{3} - 5 x^{5} + 15 x^{4} p - 5 x^{3} (3 p^{2}) - 5 x^{2} (- p^{3}) + 8 x \cdot x^{3} + 8 x (- 3 x^{2} p) + 8 x (3 x p^{2}) + 8 x (- p^{3}) - 4 x^{3} - 4 (- 3 x^{2} p) - 4 (3 x p^{2}) - 4 (- p^{3})$

Step 1.9.12

Rewrite using the commutative property of multiplication.

$x^{6} - 3 x^{5} p + 3 x^{4} p^{2} - x^{3} p^{3} - 5 x^{5} + 15 x^{4} p - 5 \cdot 3 x^{3} p^{2} - 5 x^{2} (- p^{3}) + 8 x \cdot x^{3} + 8 x (- 3 x^{2} p) + 8 x (3 x p^{2}) + 8 x (- p^{3}) - 4 x^{3} - 4 (- 3 x^{2} p) - 4 (3 x p^{2}) - 4 (- p^{3})$

Step 1.9.13

Multiply $- 5$ by $3$ .

$x^{6} - 3 x^{5} p + 3 x^{4} p^{2} - x^{3} p^{3} - 5 x^{5} + 15 x^{4} p - 15 x^{3} p^{2} - 5 x^{2} (- p^{3}) + 8 x \cdot x^{3} + 8 x (- 3 x^{2} p) + 8 x (3 x p^{2}) + 8 x (- p^{3}) - 4 x^{3} - 4 (- 3 x^{2} p) - 4 (3 x p^{2}) - 4 (- p^{3})$

Step 1.9.14

Rewrite using the commutative property of multiplication.

$x^{6} - 3 x^{5} p + 3 x^{4} p^{2} - x^{3} p^{3} - 5 x^{5} + 15 x^{4} p - 15 x^{3} p^{2} - 5 \cdot - 1 x^{2} p^{3} + 8 x \cdot x^{3} + 8 x (- 3 x^{2} p) + 8 x (3 x p^{2}) + 8 x (- p^{3}) - 4 x^{3} - 4 (- 3 x^{2} p) - 4 (3 x p^{2}) - 4 (- p^{3})$

Step 1.9.15

Multiply $- 5$ by $- 1$ .

$x^{6} - 3 x^{5} p + 3 x^{4} p^{2} - x^{3} p^{3} - 5 x^{5} + 15 x^{4} p - 15 x^{3} p^{2} + 5 x^{2} p^{3} + 8 x \cdot x^{3} + 8 x (- 3 x^{2} p) + 8 x (3 x p^{2}) + 8 x (- p^{3}) - 4 x^{3} - 4 (- 3 x^{2} p) - 4 (3 x p^{2}) - 4 (- p^{3})$

Step 1.9.16

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

Tap for more steps...

Step 1.9.16.1

Move $x^{3}$ .

$x^{6} - 3 x^{5} p + 3 x^{4} p^{2} - x^{3} p^{3} - 5 x^{5} + 15 x^{4} p - 15 x^{3} p^{2} + 5 x^{2} p^{3} + 8 (x^{3} x) + 8 x (- 3 x^{2} p) + 8 x (3 x p^{2}) + 8 x (- p^{3}) - 4 x^{3} - 4 (- 3 x^{2} p) - 4 (3 x p^{2}) - 4 (- p^{3})$

Step 1.9.16.2

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

Tap for more steps...

Step 1.9.16.2.1

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

$x^{6} - 3 x^{5} p + 3 x^{4} p^{2} - x^{3} p^{3} - 5 x^{5} + 15 x^{4} p - 15 x^{3} p^{2} + 5 x^{2} p^{3} + 8 (x^{3} x^{1}) + 8 x (- 3 x^{2} p) + 8 x (3 x p^{2}) + 8 x (- p^{3}) - 4 x^{3} - 4 (- 3 x^{2} p) - 4 (3 x p^{2}) - 4 (- p^{3})$

Step 1.9.16.2.2

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

$x^{6} - 3 x^{5} p + 3 x^{4} p^{2} - x^{3} p^{3} - 5 x^{5} + 15 x^{4} p - 15 x^{3} p^{2} + 5 x^{2} p^{3} + 8 x^{3 + 1} + 8 x (- 3 x^{2} p) + 8 x (3 x p^{2}) + 8 x (- p^{3}) - 4 x^{3} - 4 (- 3 x^{2} p) - 4 (3 x p^{2}) - 4 (- p^{3})$

Step 1.9.16.3

Add $3$ and $1$ .

$x^{6} - 3 x^{5} p + 3 x^{4} p^{2} - x^{3} p^{3} - 5 x^{5} + 15 x^{4} p - 15 x^{3} p^{2} + 5 x^{2} p^{3} + 8 x^{4} + 8 x (- 3 x^{2} p) + 8 x (3 x p^{2}) + 8 x (- p^{3}) - 4 x^{3} - 4 (- 3 x^{2} p) - 4 (3 x p^{2}) - 4 (- p^{3})$

Step 1.9.17

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

Tap for more steps...

Step 1.9.17.1

Move $x^{2}$ .

$x^{6} - 3 x^{5} p + 3 x^{4} p^{2} - x^{3} p^{3} - 5 x^{5} + 15 x^{4} p - 15 x^{3} p^{2} + 5 x^{2} p^{3} + 8 x^{4} + 8 (x^{2} x) (- 3 p) + 8 x (3 x p^{2}) + 8 x (- p^{3}) - 4 x^{3} - 4 (- 3 x^{2} p) - 4 (3 x p^{2}) - 4 (- p^{3})$

Step 1.9.17.2

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

Tap for more steps...

Step 1.9.17.2.1

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

$x^{6} - 3 x^{5} p + 3 x^{4} p^{2} - x^{3} p^{3} - 5 x^{5} + 15 x^{4} p - 15 x^{3} p^{2} + 5 x^{2} p^{3} + 8 x^{4} + 8 (x^{2} x^{1}) (- 3 p) + 8 x (3 x p^{2}) + 8 x (- p^{3}) - 4 x^{3} - 4 (- 3 x^{2} p) - 4 (3 x p^{2}) - 4 (- p^{3})$

Step 1.9.17.2.2

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

$x^{6} - 3 x^{5} p + 3 x^{4} p^{2} - x^{3} p^{3} - 5 x^{5} + 15 x^{4} p - 15 x^{3} p^{2} + 5 x^{2} p^{3} + 8 x^{4} + 8 x^{2 + 1} (- 3 p) + 8 x (3 x p^{2}) + 8 x (- p^{3}) - 4 x^{3} - 4 (- 3 x^{2} p) - 4 (3 x p^{2}) - 4 (- p^{3})$

Step 1.9.17.3

Add $2$ and $1$ .

$x^{6} - 3 x^{5} p + 3 x^{4} p^{2} - x^{3} p^{3} - 5 x^{5} + 15 x^{4} p - 15 x^{3} p^{2} + 5 x^{2} p^{3} + 8 x^{4} + 8 x^{3} (- 3 p) + 8 x (3 x p^{2}) + 8 x (- p^{3}) - 4 x^{3} - 4 (- 3 x^{2} p) - 4 (3 x p^{2}) - 4 (- p^{3})$

Step 1.9.18

Rewrite using the commutative property of multiplication.

$x^{6} - 3 x^{5} p + 3 x^{4} p^{2} - x^{3} p^{3} - 5 x^{5} + 15 x^{4} p - 15 x^{3} p^{2} + 5 x^{2} p^{3} + 8 x^{4} + 8 \cdot - 3 x^{3} p + 8 x (3 x p^{2}) + 8 x (- p^{3}) - 4 x^{3} - 4 (- 3 x^{2} p) - 4 (3 x p^{2}) - 4 (- p^{3})$

Step 1.9.19

Multiply $8$ by $- 3$ .

$x^{6} - 3 x^{5} p + 3 x^{4} p^{2} - x^{3} p^{3} - 5 x^{5} + 15 x^{4} p - 15 x^{3} p^{2} + 5 x^{2} p^{3} + 8 x^{4} - 24 x^{3} p + 8 x (3 x p^{2}) + 8 x (- p^{3}) - 4 x^{3} - 4 (- 3 x^{2} p) - 4 (3 x p^{2}) - 4 (- p^{3})$

Step 1.9.20

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

Tap for more steps...

Step 1.9.20.1

Move $x$ .

$x^{6} - 3 x^{5} p + 3 x^{4} p^{2} - x^{3} p^{3} - 5 x^{5} + 15 x^{4} p - 15 x^{3} p^{2} + 5 x^{2} p^{3} + 8 x^{4} - 24 x^{3} p + 8 (x \cdot x) (3 p^{2}) + 8 x (- p^{3}) - 4 x^{3} - 4 (- 3 x^{2} p) - 4 (3 x p^{2}) - 4 (- p^{3})$

Step 1.9.20.2

Multiply $x$ by $x$ .

$x^{6} - 3 x^{5} p + 3 x^{4} p^{2} - x^{3} p^{3} - 5 x^{5} + 15 x^{4} p - 15 x^{3} p^{2} + 5 x^{2} p^{3} + 8 x^{4} - 24 x^{3} p + 8 x^{2} (3 p^{2}) + 8 x (- p^{3}) - 4 x^{3} - 4 (- 3 x^{2} p) - 4 (3 x p^{2}) - 4 (- p^{3})$

Step 1.9.21

Rewrite using the commutative property of multiplication.

$x^{6} - 3 x^{5} p + 3 x^{4} p^{2} - x^{3} p^{3} - 5 x^{5} + 15 x^{4} p - 15 x^{3} p^{2} + 5 x^{2} p^{3} + 8 x^{4} - 24 x^{3} p + 8 \cdot 3 x^{2} p^{2} + 8 x (- p^{3}) - 4 x^{3} - 4 (- 3 x^{2} p) - 4 (3 x p^{2}) - 4 (- p^{3})$

Step 1.9.22

Multiply $8$ by $3$ .

$x^{6} - 3 x^{5} p + 3 x^{4} p^{2} - x^{3} p^{3} - 5 x^{5} + 15 x^{4} p - 15 x^{3} p^{2} + 5 x^{2} p^{3} + 8 x^{4} - 24 x^{3} p + 24 x^{2} p^{2} + 8 x (- p^{3}) - 4 x^{3} - 4 (- 3 x^{2} p) - 4 (3 x p^{2}) - 4 (- p^{3})$

Step 1.9.23

Rewrite using the commutative property of multiplication.

$x^{6} - 3 x^{5} p + 3 x^{4} p^{2} - x^{3} p^{3} - 5 x^{5} + 15 x^{4} p - 15 x^{3} p^{2} + 5 x^{2} p^{3} + 8 x^{4} - 24 x^{3} p + 24 x^{2} p^{2} + 8 \cdot - 1 x p^{3} - 4 x^{3} - 4 (- 3 x^{2} p) - 4 (3 x p^{2}) - 4 (- p^{3})$

Step 1.9.24

Multiply $8$ by $- 1$ .

$x^{6} - 3 x^{5} p + 3 x^{4} p^{2} - x^{3} p^{3} - 5 x^{5} + 15 x^{4} p - 15 x^{3} p^{2} + 5 x^{2} p^{3} + 8 x^{4} - 24 x^{3} p + 24 x^{2} p^{2} - 8 x p^{3} - 4 x^{3} - 4 (- 3 x^{2} p) - 4 (3 x p^{2}) - 4 (- p^{3})$

Step 1.9.25

Multiply $- 3$ by $- 4$ .

$x^{6} - 3 x^{5} p + 3 x^{4} p^{2} - x^{3} p^{3} - 5 x^{5} + 15 x^{4} p - 15 x^{3} p^{2} + 5 x^{2} p^{3} + 8 x^{4} - 24 x^{3} p + 24 x^{2} p^{2} - 8 x p^{3} - 4 x^{3} + 12 (x^{2} p) - 4 (3 x p^{2}) - 4 (- p^{3})$

Step 1.9.26

Multiply $3$ by $- 4$ .

$x^{6} - 3 x^{5} p + 3 x^{4} p^{2} - x^{3} p^{3} - 5 x^{5} + 15 x^{4} p - 15 x^{3} p^{2} + 5 x^{2} p^{3} + 8 x^{4} - 24 x^{3} p + 24 x^{2} p^{2} - 8 x p^{3} - 4 x^{3} + 12 x^{2} p - 12 (x p^{2}) - 4 (- p^{3})$

Step 1.9.27

Multiply $- 1$ by $- 4$ .

$x^{6} - 3 x^{5} p + 3 x^{4} p^{2} - x^{3} p^{3} - 5 x^{5} + 15 x^{4} p - 15 x^{3} p^{2} + 5 x^{2} p^{3} + 8 x^{4} - 24 x^{3} p + 24 x^{2} p^{2} - 8 x p^{3} - 4 x^{3} + 12 x^{2} p - 12 x p^{2} + 4 p^{3}$

Step 2

The largest exponent is the degree of the polynomial.

$6$