Find the Degree f(x)=(x+5)^3(x-9)(x+1)

$f (x) = {(x + 5)}^{3} (x - 9) (x + 1)$

Step 1

Simplify and reorder the polynomial.

Use the Binomial Theorem.

$(x^{3} + 3 x^{2} \cdot 5 + 3 x \cdot 5^{2} + 5^{3}) (x - 9) (x + 1)$

Simplify each term.

Multiply $5$ by $3$ .

$(x^{3} + 15 x^{2} + 3 x \cdot 5^{2} + 5^{3}) (x - 9) (x + 1)$

Raise $5$ to the power of $2$ .

$(x^{3} + 15 x^{2} + 3 x \cdot 25 + 5^{3}) (x - 9) (x + 1)$

Multiply $25$ by $3$ .

$(x^{3} + 15 x^{2} + 75 x + 5^{3}) (x - 9) (x + 1)$

Raise $5$ to the power of $3$ .

$(x^{3} + 15 x^{2} + 75 x + 125) (x - 9) (x + 1)$

Expand $(x^{3} + 15 x^{2} + 75 x + 125) (x - 9)$ by multiplying each term in the first expression by each term in the second expression.

$(x^{3} x + x^{3} \cdot - 9 + 15 x^{2} x + 15 x^{2} \cdot - 9 + 75 x \cdot x + 75 x \cdot - 9 + 125 x + 125 \cdot - 9) (x + 1)$

Simplify terms.

Simplify each term.

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

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

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

$(x^{3} x^{1} + x^{3} \cdot - 9 + 15 x^{2} x + 15 x^{2} \cdot - 9 + 75 x \cdot x + 75 x \cdot - 9 + 125 x + 125 \cdot - 9) (x + 1)$

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

$(x^{3 + 1} + x^{3} \cdot - 9 + 15 x^{2} x + 15 x^{2} \cdot - 9 + 75 x \cdot x + 75 x \cdot - 9 + 125 x + 125 \cdot - 9) (x + 1)$

Add $3$ and $1$ .

$(x^{4} + x^{3} \cdot - 9 + 15 x^{2} x + 15 x^{2} \cdot - 9 + 75 x \cdot x + 75 x \cdot - 9 + 125 x + 125 \cdot - 9) (x + 1)$

Move $- 9$ to the left of $x^{3}$ .

$(x^{4} - 9 \cdot x^{3} + 15 x^{2} x + 15 x^{2} \cdot - 9 + 75 x \cdot x + 75 x \cdot - 9 + 125 x + 125 \cdot - 9) (x + 1)$

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

Move $x$ .

$(x^{4} - 9 x^{3} + 15 (x \cdot x^{2}) + 15 x^{2} \cdot - 9 + 75 x \cdot x + 75 x \cdot - 9 + 125 x + 125 \cdot - 9) (x + 1)$

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

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

$(x^{4} - 9 x^{3} + 15 (x^{1} x^{2}) + 15 x^{2} \cdot - 9 + 75 x \cdot x + 75 x \cdot - 9 + 125 x + 125 \cdot - 9) (x + 1)$

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

$(x^{4} - 9 x^{3} + 15 x^{1 + 2} + 15 x^{2} \cdot - 9 + 75 x \cdot x + 75 x \cdot - 9 + 125 x + 125 \cdot - 9) (x + 1)$

Add $1$ and $2$ .

$(x^{4} - 9 x^{3} + 15 x^{3} + 15 x^{2} \cdot - 9 + 75 x \cdot x + 75 x \cdot - 9 + 125 x + 125 \cdot - 9) (x + 1)$

Multiply $- 9$ by $15$ .

$(x^{4} - 9 x^{3} + 15 x^{3} - 135 x^{2} + 75 x \cdot x + 75 x \cdot - 9 + 125 x + 125 \cdot - 9) (x + 1)$

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

Move $x$ .

$(x^{4} - 9 x^{3} + 15 x^{3} - 135 x^{2} + 75 (x \cdot x) + 75 x \cdot - 9 + 125 x + 125 \cdot - 9) (x + 1)$

Multiply $x$ by $x$ .

$(x^{4} - 9 x^{3} + 15 x^{3} - 135 x^{2} + 75 x^{2} + 75 x \cdot - 9 + 125 x + 125 \cdot - 9) (x + 1)$

Multiply $- 9$ by $75$ .

$(x^{4} - 9 x^{3} + 15 x^{3} - 135 x^{2} + 75 x^{2} - 675 x + 125 x + 125 \cdot - 9) (x + 1)$

Multiply $125$ by $- 9$ .

$(x^{4} - 9 x^{3} + 15 x^{3} - 135 x^{2} + 75 x^{2} - 675 x + 125 x - 1125) (x + 1)$

Simplify by adding terms.

Add $- 9 x^{3}$ and $15 x^{3}$ .

$(x^{4} + 6 x^{3} - 135 x^{2} + 75 x^{2} - 675 x + 125 x - 1125) (x + 1)$

Add $- 135 x^{2}$ and $75 x^{2}$ .

$(x^{4} + 6 x^{3} - 60 x^{2} - 675 x + 125 x - 1125) (x + 1)$

Add $- 675 x$ and $125 x$ .

$(x^{4} + 6 x^{3} - 60 x^{2} - 550 x - 1125) (x + 1)$

Expand $(x^{4} + 6 x^{3} - 60 x^{2} - 550 x - 1125) (x + 1)$ by multiplying each term in the first expression by each term in the second expression.

$x^{4} x + x^{4} \cdot 1 + 6 x^{3} x + 6 x^{3} \cdot 1 - 60 x^{2} x - 60 x^{2} \cdot 1 - 550 x \cdot x - 550 x \cdot 1 - 1125 x - 1125 \cdot 1$

Simplify terms.

Simplify each term.

Multiply $x^{4}$ by $x$ by adding the exponents.

Multiply $x^{4}$ by $x$ .

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

$x^{4} x^{1} + x^{4} \cdot 1 + 6 x^{3} x + 6 x^{3} \cdot 1 - 60 x^{2} x - 60 x^{2} \cdot 1 - 550 x \cdot x - 550 x \cdot 1 - 1125 x - 1125 \cdot 1$

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

$x^{4 + 1} + x^{4} \cdot 1 + 6 x^{3} x + 6 x^{3} \cdot 1 - 60 x^{2} x - 60 x^{2} \cdot 1 - 550 x \cdot x - 550 x \cdot 1 - 1125 x - 1125 \cdot 1$

Add $4$ and $1$ .

$x^{5} + x^{4} \cdot 1 + 6 x^{3} x + 6 x^{3} \cdot 1 - 60 x^{2} x - 60 x^{2} \cdot 1 - 550 x \cdot x - 550 x \cdot 1 - 1125 x - 1125 \cdot 1$

Multiply $x^{4}$ by $1$ .

$x^{5} + x^{4} + 6 x^{3} x + 6 x^{3} \cdot 1 - 60 x^{2} x - 60 x^{2} \cdot 1 - 550 x \cdot x - 550 x \cdot 1 - 1125 x - 1125 \cdot 1$

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

Move $x$ .

$x^{5} + x^{4} + 6 (x \cdot x^{3}) + 6 x^{3} \cdot 1 - 60 x^{2} x - 60 x^{2} \cdot 1 - 550 x \cdot x - 550 x \cdot 1 - 1125 x - 1125 \cdot 1$

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

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

$x^{5} + x^{4} + 6 (x^{1} x^{3}) + 6 x^{3} \cdot 1 - 60 x^{2} x - 60 x^{2} \cdot 1 - 550 x \cdot x - 550 x \cdot 1 - 1125 x - 1125 \cdot 1$

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

$x^{5} + x^{4} + 6 x^{1 + 3} + 6 x^{3} \cdot 1 - 60 x^{2} x - 60 x^{2} \cdot 1 - 550 x \cdot x - 550 x \cdot 1 - 1125 x - 1125 \cdot 1$

Add $1$ and $3$ .

$x^{5} + x^{4} + 6 x^{4} + 6 x^{3} \cdot 1 - 60 x^{2} x - 60 x^{2} \cdot 1 - 550 x \cdot x - 550 x \cdot 1 - 1125 x - 1125 \cdot 1$

Multiply $6$ by $1$ .

$x^{5} + x^{4} + 6 x^{4} + 6 x^{3} - 60 x^{2} x - 60 x^{2} \cdot 1 - 550 x \cdot x - 550 x \cdot 1 - 1125 x - 1125 \cdot 1$

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

Move $x$ .

$x^{5} + x^{4} + 6 x^{4} + 6 x^{3} - 60 (x \cdot x^{2}) - 60 x^{2} \cdot 1 - 550 x \cdot x - 550 x \cdot 1 - 1125 x - 1125 \cdot 1$

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

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

$x^{5} + x^{4} + 6 x^{4} + 6 x^{3} - 60 (x^{1} x^{2}) - 60 x^{2} \cdot 1 - 550 x \cdot x - 550 x \cdot 1 - 1125 x - 1125 \cdot 1$

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

$x^{5} + x^{4} + 6 x^{4} + 6 x^{3} - 60 x^{1 + 2} - 60 x^{2} \cdot 1 - 550 x \cdot x - 550 x \cdot 1 - 1125 x - 1125 \cdot 1$

Add $1$ and $2$ .

$x^{5} + x^{4} + 6 x^{4} + 6 x^{3} - 60 x^{3} - 60 x^{2} \cdot 1 - 550 x \cdot x - 550 x \cdot 1 - 1125 x - 1125 \cdot 1$

Multiply $- 60$ by $1$ .

$x^{5} + x^{4} + 6 x^{4} + 6 x^{3} - 60 x^{3} - 60 x^{2} - 550 x \cdot x - 550 x \cdot 1 - 1125 x - 1125 \cdot 1$

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

Move $x$ .

$x^{5} + x^{4} + 6 x^{4} + 6 x^{3} - 60 x^{3} - 60 x^{2} - 550 (x \cdot x) - 550 x \cdot 1 - 1125 x - 1125 \cdot 1$

Multiply $x$ by $x$ .

$x^{5} + x^{4} + 6 x^{4} + 6 x^{3} - 60 x^{3} - 60 x^{2} - 550 x^{2} - 550 x \cdot 1 - 1125 x - 1125 \cdot 1$

Multiply $- 550$ by $1$ .

$x^{5} + x^{4} + 6 x^{4} + 6 x^{3} - 60 x^{3} - 60 x^{2} - 550 x^{2} - 550 x - 1125 x - 1125 \cdot 1$

Multiply $- 1125$ by $1$ .

$x^{5} + x^{4} + 6 x^{4} + 6 x^{3} - 60 x^{3} - 60 x^{2} - 550 x^{2} - 550 x - 1125 x - 1125$

Simplify by adding terms.

Add $x^{4}$ and $6 x^{4}$ .

$x^{5} + 7 x^{4} + 6 x^{3} - 60 x^{3} - 60 x^{2} - 550 x^{2} - 550 x - 1125 x - 1125$

Subtract $60 x^{3}$ from $6 x^{3}$ .

$x^{5} + 7 x^{4} - 54 x^{3} - 60 x^{2} - 550 x^{2} - 550 x - 1125 x - 1125$

Subtract $550 x^{2}$ from $- 60 x^{2}$ .

$x^{5} + 7 x^{4} - 54 x^{3} - 610 x^{2} - 550 x - 1125 x - 1125$

Subtract $1125 x$ from $- 550 x$ .

$x^{5} + 7 x^{4} - 54 x^{3} - 610 x^{2} - 1675 x - 1125$

Step 2

The largest exponent is the degree of the polynomial.

$5$

Find the Degree f(x)=(x+5)^3(x-9)(x+1)

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