Find the Degree (x+4)^2(x-1)^3=0

${(x + 4)}^{2} {(x - 1)}^{3} = 0$

Step 1

Simplify and reorder the polynomial.

Rewrite ${(x + 4)}^{2}$ as $(x + 4) (x + 4)$ .

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

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

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

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

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

Multiply $4$ by $4$ .

$(x^{2} + 4 x + 4 x + 16) {(x - 1)}^{3}$

Add $4 x$ and $4 x$ .

$(x^{2} + 8 x + 16) {(x - 1)}^{3}$

Use the Binomial Theorem.

$(x^{2} + 8 x + 16) (x^{3} + 3 x^{2} \cdot - 1 + 3 x {(- 1)}^{2} + {(- 1)}^{3})$

Simplify each term.

Multiply $- 1$ by $3$ .

$(x^{2} + 8 x + 16) (x^{3} - 3 x^{2} + 3 x {(- 1)}^{2} + {(- 1)}^{3})$

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

$(x^{2} + 8 x + 16) (x^{3} - 3 x^{2} + 3 x \cdot 1 + {(- 1)}^{3})$

Multiply $3$ by $1$ .

$(x^{2} + 8 x + 16) (x^{3} - 3 x^{2} + 3 x + {(- 1)}^{3})$

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

$(x^{2} + 8 x + 16) (x^{3} - 3 x^{2} + 3 x - 1)$

Expand $(x^{2} + 8 x + 16) (x^{3} - 3 x^{2} + 3 x - 1)$ by multiplying each term in the first expression by each term in the second expression.

$x^{2} x^{3} + x^{2} (- 3 x^{2}) + x^{2} (3 x) + x^{2} \cdot - 1 + 8 x \cdot x^{3} + 8 x (- 3 x^{2}) + 8 x (3 x) + 8 x \cdot - 1 + 16 x^{3} + 16 (- 3 x^{2}) + 16 (3 x) + 16 \cdot - 1$

Simplify terms.

Simplify each term.

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

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

$x^{2 + 3} + x^{2} (- 3 x^{2}) + x^{2} (3 x) + x^{2} \cdot - 1 + 8 x \cdot x^{3} + 8 x (- 3 x^{2}) + 8 x (3 x) + 8 x \cdot - 1 + 16 x^{3} + 16 (- 3 x^{2}) + 16 (3 x) + 16 \cdot - 1$

Add $2$ and $3$ .

$x^{5} + x^{2} (- 3 x^{2}) + x^{2} (3 x) + x^{2} \cdot - 1 + 8 x \cdot x^{3} + 8 x (- 3 x^{2}) + 8 x (3 x) + 8 x \cdot - 1 + 16 x^{3} + 16 (- 3 x^{2}) + 16 (3 x) + 16 \cdot - 1$

Rewrite using the commutative property of multiplication.

$x^{5} - 3 x^{2} x^{2} + x^{2} (3 x) + x^{2} \cdot - 1 + 8 x \cdot x^{3} + 8 x (- 3 x^{2}) + 8 x (3 x) + 8 x \cdot - 1 + 16 x^{3} + 16 (- 3 x^{2}) + 16 (3 x) + 16 \cdot - 1$

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

Move $x^{2}$ .

$x^{5} - 3 (x^{2} x^{2}) + x^{2} (3 x) + x^{2} \cdot - 1 + 8 x \cdot x^{3} + 8 x (- 3 x^{2}) + 8 x (3 x) + 8 x \cdot - 1 + 16 x^{3} + 16 (- 3 x^{2}) + 16 (3 x) + 16 \cdot - 1$

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

$x^{5} - 3 x^{2 + 2} + x^{2} (3 x) + x^{2} \cdot - 1 + 8 x \cdot x^{3} + 8 x (- 3 x^{2}) + 8 x (3 x) + 8 x \cdot - 1 + 16 x^{3} + 16 (- 3 x^{2}) + 16 (3 x) + 16 \cdot - 1$

Add $2$ and $2$ .

$x^{5} - 3 x^{4} + x^{2} (3 x) + x^{2} \cdot - 1 + 8 x \cdot x^{3} + 8 x (- 3 x^{2}) + 8 x (3 x) + 8 x \cdot - 1 + 16 x^{3} + 16 (- 3 x^{2}) + 16 (3 x) + 16 \cdot - 1$

Rewrite using the commutative property of multiplication.

$x^{5} - 3 x^{4} + 3 x^{2} x + x^{2} \cdot - 1 + 8 x \cdot x^{3} + 8 x (- 3 x^{2}) + 8 x (3 x) + 8 x \cdot - 1 + 16 x^{3} + 16 (- 3 x^{2}) + 16 (3 x) + 16 \cdot - 1$

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

Move $x$ .

$x^{5} - 3 x^{4} + 3 (x \cdot x^{2}) + x^{2} \cdot - 1 + 8 x \cdot x^{3} + 8 x (- 3 x^{2}) + 8 x (3 x) + 8 x \cdot - 1 + 16 x^{3} + 16 (- 3 x^{2}) + 16 (3 x) + 16 \cdot - 1$

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

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

$x^{5} - 3 x^{4} + 3 (x^{1} x^{2}) + x^{2} \cdot - 1 + 8 x \cdot x^{3} + 8 x (- 3 x^{2}) + 8 x (3 x) + 8 x \cdot - 1 + 16 x^{3} + 16 (- 3 x^{2}) + 16 (3 x) + 16 \cdot - 1$

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

$x^{5} - 3 x^{4} + 3 x^{1 + 2} + x^{2} \cdot - 1 + 8 x \cdot x^{3} + 8 x (- 3 x^{2}) + 8 x (3 x) + 8 x \cdot - 1 + 16 x^{3} + 16 (- 3 x^{2}) + 16 (3 x) + 16 \cdot - 1$

Add $1$ and $2$ .

$x^{5} - 3 x^{4} + 3 x^{3} + x^{2} \cdot - 1 + 8 x \cdot x^{3} + 8 x (- 3 x^{2}) + 8 x (3 x) + 8 x \cdot - 1 + 16 x^{3} + 16 (- 3 x^{2}) + 16 (3 x) + 16 \cdot - 1$

Move $- 1$ to the left of $x^{2}$ .

$x^{5} - 3 x^{4} + 3 x^{3} - 1 \cdot x^{2} + 8 x \cdot x^{3} + 8 x (- 3 x^{2}) + 8 x (3 x) + 8 x \cdot - 1 + 16 x^{3} + 16 (- 3 x^{2}) + 16 (3 x) + 16 \cdot - 1$

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

$x^{5} - 3 x^{4} + 3 x^{3} - x^{2} + 8 x \cdot x^{3} + 8 x (- 3 x^{2}) + 8 x (3 x) + 8 x \cdot - 1 + 16 x^{3} + 16 (- 3 x^{2}) + 16 (3 x) + 16 \cdot - 1$

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

Move $x^{3}$ .

$x^{5} - 3 x^{4} + 3 x^{3} - x^{2} + 8 (x^{3} x) + 8 x (- 3 x^{2}) + 8 x (3 x) + 8 x \cdot - 1 + 16 x^{3} + 16 (- 3 x^{2}) + 16 (3 x) + 16 \cdot - 1$

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

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

$x^{5} - 3 x^{4} + 3 x^{3} - x^{2} + 8 (x^{3} x^{1}) + 8 x (- 3 x^{2}) + 8 x (3 x) + 8 x \cdot - 1 + 16 x^{3} + 16 (- 3 x^{2}) + 16 (3 x) + 16 \cdot - 1$

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

$x^{5} - 3 x^{4} + 3 x^{3} - x^{2} + 8 x^{3 + 1} + 8 x (- 3 x^{2}) + 8 x (3 x) + 8 x \cdot - 1 + 16 x^{3} + 16 (- 3 x^{2}) + 16 (3 x) + 16 \cdot - 1$

Add $3$ and $1$ .

$x^{5} - 3 x^{4} + 3 x^{3} - x^{2} + 8 x^{4} + 8 x (- 3 x^{2}) + 8 x (3 x) + 8 x \cdot - 1 + 16 x^{3} + 16 (- 3 x^{2}) + 16 (3 x) + 16 \cdot - 1$

Rewrite using the commutative property of multiplication.

$x^{5} - 3 x^{4} + 3 x^{3} - x^{2} + 8 x^{4} + 8 \cdot - 3 x \cdot x^{2} + 8 x (3 x) + 8 x \cdot - 1 + 16 x^{3} + 16 (- 3 x^{2}) + 16 (3 x) + 16 \cdot - 1$

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

Move $x^{2}$ .

$x^{5} - 3 x^{4} + 3 x^{3} - x^{2} + 8 x^{4} + 8 \cdot - 3 (x^{2} x) + 8 x (3 x) + 8 x \cdot - 1 + 16 x^{3} + 16 (- 3 x^{2}) + 16 (3 x) + 16 \cdot - 1$

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

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

$x^{5} - 3 x^{4} + 3 x^{3} - x^{2} + 8 x^{4} + 8 \cdot - 3 (x^{2} x^{1}) + 8 x (3 x) + 8 x \cdot - 1 + 16 x^{3} + 16 (- 3 x^{2}) + 16 (3 x) + 16 \cdot - 1$

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

$x^{5} - 3 x^{4} + 3 x^{3} - x^{2} + 8 x^{4} + 8 \cdot - 3 x^{2 + 1} + 8 x (3 x) + 8 x \cdot - 1 + 16 x^{3} + 16 (- 3 x^{2}) + 16 (3 x) + 16 \cdot - 1$

Add $2$ and $1$ .

$x^{5} - 3 x^{4} + 3 x^{3} - x^{2} + 8 x^{4} + 8 \cdot - 3 x^{3} + 8 x (3 x) + 8 x \cdot - 1 + 16 x^{3} + 16 (- 3 x^{2}) + 16 (3 x) + 16 \cdot - 1$

Multiply $8$ by $- 3$ .

$x^{5} - 3 x^{4} + 3 x^{3} - x^{2} + 8 x^{4} - 24 x^{3} + 8 x (3 x) + 8 x \cdot - 1 + 16 x^{3} + 16 (- 3 x^{2}) + 16 (3 x) + 16 \cdot - 1$

Rewrite using the commutative property of multiplication.

$x^{5} - 3 x^{4} + 3 x^{3} - x^{2} + 8 x^{4} - 24 x^{3} + 8 \cdot 3 x \cdot x + 8 x \cdot - 1 + 16 x^{3} + 16 (- 3 x^{2}) + 16 (3 x) + 16 \cdot - 1$

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

Move $x$ .

$x^{5} - 3 x^{4} + 3 x^{3} - x^{2} + 8 x^{4} - 24 x^{3} + 8 \cdot 3 (x \cdot x) + 8 x \cdot - 1 + 16 x^{3} + 16 (- 3 x^{2}) + 16 (3 x) + 16 \cdot - 1$

Multiply $x$ by $x$ .

$x^{5} - 3 x^{4} + 3 x^{3} - x^{2} + 8 x^{4} - 24 x^{3} + 8 \cdot 3 x^{2} + 8 x \cdot - 1 + 16 x^{3} + 16 (- 3 x^{2}) + 16 (3 x) + 16 \cdot - 1$

Multiply $8$ by $3$ .

$x^{5} - 3 x^{4} + 3 x^{3} - x^{2} + 8 x^{4} - 24 x^{3} + 24 x^{2} + 8 x \cdot - 1 + 16 x^{3} + 16 (- 3 x^{2}) + 16 (3 x) + 16 \cdot - 1$

Multiply $- 1$ by $8$ .

$x^{5} - 3 x^{4} + 3 x^{3} - x^{2} + 8 x^{4} - 24 x^{3} + 24 x^{2} - 8 x + 16 x^{3} + 16 (- 3 x^{2}) + 16 (3 x) + 16 \cdot - 1$

Multiply $- 3$ by $16$ .

$x^{5} - 3 x^{4} + 3 x^{3} - x^{2} + 8 x^{4} - 24 x^{3} + 24 x^{2} - 8 x + 16 x^{3} - 48 x^{2} + 16 (3 x) + 16 \cdot - 1$

Multiply $3$ by $16$ .

$x^{5} - 3 x^{4} + 3 x^{3} - x^{2} + 8 x^{4} - 24 x^{3} + 24 x^{2} - 8 x + 16 x^{3} - 48 x^{2} + 48 x + 16 \cdot - 1$

Multiply $16$ by $- 1$ .

$x^{5} - 3 x^{4} + 3 x^{3} - x^{2} + 8 x^{4} - 24 x^{3} + 24 x^{2} - 8 x + 16 x^{3} - 48 x^{2} + 48 x - 16$

Simplify by adding terms.

Add $- 3 x^{4}$ and $8 x^{4}$ .

$x^{5} + 5 x^{4} + 3 x^{3} - x^{2} - 24 x^{3} + 24 x^{2} - 8 x + 16 x^{3} - 48 x^{2} + 48 x - 16$

Subtract $24 x^{3}$ from $3 x^{3}$ .

$x^{5} + 5 x^{4} - 21 x^{3} - x^{2} + 24 x^{2} - 8 x + 16 x^{3} - 48 x^{2} + 48 x - 16$

Add $- 21 x^{3}$ and $16 x^{3}$ .

$x^{5} + 5 x^{4} - 5 x^{3} - x^{2} + 24 x^{2} - 8 x - 48 x^{2} + 48 x - 16$

Add $- x^{2}$ and $24 x^{2}$ .

$x^{5} + 5 x^{4} - 5 x^{3} + 23 x^{2} - 8 x - 48 x^{2} + 48 x - 16$

Subtract $48 x^{2}$ from $23 x^{2}$ .

$x^{5} + 5 x^{4} - 5 x^{3} - 25 x^{2} - 8 x + 48 x - 16$

Add $- 8 x$ and $48 x$ .

$x^{5} + 5 x^{4} - 5 x^{3} - 25 x^{2} + 40 x - 16$

Step 2

The largest exponent is the degree of the polynomial.

$5$

Find the Degree (x+4)^2(x-1)^3=0

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