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

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

Step 1

Simplify and reorder the polynomial.

Use the Binomial Theorem.

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

Simplify each term.

Multiply $- 1$ by $5$ .

$(x + 3) (x^{5} - 5 x^{4} + 10 x^{3} {(- 1)}^{2} + 10 x^{2} {(- 1)}^{3} + 5 x {(- 1)}^{4} + {(- 1)}^{5})$

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

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

Multiply $10$ by $1$ .

$(x + 3) (x^{5} - 5 x^{4} + 10 x^{3} + 10 x^{2} {(- 1)}^{3} + 5 x {(- 1)}^{4} + {(- 1)}^{5})$

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

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

Multiply $- 1$ by $10$ .

$(x + 3) (x^{5} - 5 x^{4} + 10 x^{3} - 10 x^{2} + 5 x {(- 1)}^{4} + {(- 1)}^{5})$

Raise $- 1$ to the power of $4$ .

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

Multiply $5$ by $1$ .

$(x + 3) (x^{5} - 5 x^{4} + 10 x^{3} - 10 x^{2} + 5 x + {(- 1)}^{5})$

Raise $- 1$ to the power of $5$ .

$(x + 3) (x^{5} - 5 x^{4} + 10 x^{3} - 10 x^{2} + 5 x - 1)$

Expand $(x + 3) (x^{5} - 5 x^{4} + 10 x^{3} - 10 x^{2} + 5 x - 1)$ by multiplying each term in the first expression by each term in the second expression.

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

Simplify terms.

Simplify each term.

Multiply $x$ by $x^{5}$ by adding the exponents.

Multiply $x$ by $x^{5}$ .

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

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

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

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

Add $1$ and $5$ .

$x^{6} + x (- 5 x^{4}) + x (10 x^{3}) + x (- 10 x^{2}) + x (5 x) + x \cdot - 1 + 3 x^{5} + 3 (- 5 x^{4}) + 3 (10 x^{3}) + 3 (- 10 x^{2}) + 3 (5 x) + 3 \cdot - 1$

Rewrite using the commutative property of multiplication.

$x^{6} - 5 x \cdot x^{4} + x (10 x^{3}) + x (- 10 x^{2}) + x (5 x) + x \cdot - 1 + 3 x^{5} + 3 (- 5 x^{4}) + 3 (10 x^{3}) + 3 (- 10 x^{2}) + 3 (5 x) + 3 \cdot - 1$

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

Move $x^{4}$ .

$x^{6} - 5 (x^{4} x) + x (10 x^{3}) + x (- 10 x^{2}) + x (5 x) + x \cdot - 1 + 3 x^{5} + 3 (- 5 x^{4}) + 3 (10 x^{3}) + 3 (- 10 x^{2}) + 3 (5 x) + 3 \cdot - 1$

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

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

$x^{6} - 5 (x^{4} x^{1}) + x (10 x^{3}) + x (- 10 x^{2}) + x (5 x) + x \cdot - 1 + 3 x^{5} + 3 (- 5 x^{4}) + 3 (10 x^{3}) + 3 (- 10 x^{2}) + 3 (5 x) + 3 \cdot - 1$

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

$x^{6} - 5 x^{4 + 1} + x (10 x^{3}) + x (- 10 x^{2}) + x (5 x) + x \cdot - 1 + 3 x^{5} + 3 (- 5 x^{4}) + 3 (10 x^{3}) + 3 (- 10 x^{2}) + 3 (5 x) + 3 \cdot - 1$

Add $4$ and $1$ .

$x^{6} - 5 x^{5} + x (10 x^{3}) + x (- 10 x^{2}) + x (5 x) + x \cdot - 1 + 3 x^{5} + 3 (- 5 x^{4}) + 3 (10 x^{3}) + 3 (- 10 x^{2}) + 3 (5 x) + 3 \cdot - 1$

Rewrite using the commutative property of multiplication.

$x^{6} - 5 x^{5} + 10 x \cdot x^{3} + x (- 10 x^{2}) + x (5 x) + x \cdot - 1 + 3 x^{5} + 3 (- 5 x^{4}) + 3 (10 x^{3}) + 3 (- 10 x^{2}) + 3 (5 x) + 3 \cdot - 1$

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

Move $x^{3}$ .

$x^{6} - 5 x^{5} + 10 (x^{3} x) + x (- 10 x^{2}) + x (5 x) + x \cdot - 1 + 3 x^{5} + 3 (- 5 x^{4}) + 3 (10 x^{3}) + 3 (- 10 x^{2}) + 3 (5 x) + 3 \cdot - 1$

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

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

$x^{6} - 5 x^{5} + 10 (x^{3} x^{1}) + x (- 10 x^{2}) + x (5 x) + x \cdot - 1 + 3 x^{5} + 3 (- 5 x^{4}) + 3 (10 x^{3}) + 3 (- 10 x^{2}) + 3 (5 x) + 3 \cdot - 1$

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

$x^{6} - 5 x^{5} + 10 x^{3 + 1} + x (- 10 x^{2}) + x (5 x) + x \cdot - 1 + 3 x^{5} + 3 (- 5 x^{4}) + 3 (10 x^{3}) + 3 (- 10 x^{2}) + 3 (5 x) + 3 \cdot - 1$

Add $3$ and $1$ .

$x^{6} - 5 x^{5} + 10 x^{4} + x (- 10 x^{2}) + x (5 x) + x \cdot - 1 + 3 x^{5} + 3 (- 5 x^{4}) + 3 (10 x^{3}) + 3 (- 10 x^{2}) + 3 (5 x) + 3 \cdot - 1$

Rewrite using the commutative property of multiplication.

$x^{6} - 5 x^{5} + 10 x^{4} - 10 x \cdot x^{2} + x (5 x) + x \cdot - 1 + 3 x^{5} + 3 (- 5 x^{4}) + 3 (10 x^{3}) + 3 (- 10 x^{2}) + 3 (5 x) + 3 \cdot - 1$

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

Move $x^{2}$ .

$x^{6} - 5 x^{5} + 10 x^{4} - 10 (x^{2} x) + x (5 x) + x \cdot - 1 + 3 x^{5} + 3 (- 5 x^{4}) + 3 (10 x^{3}) + 3 (- 10 x^{2}) + 3 (5 x) + 3 \cdot - 1$

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

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

$x^{6} - 5 x^{5} + 10 x^{4} - 10 (x^{2} x^{1}) + x (5 x) + x \cdot - 1 + 3 x^{5} + 3 (- 5 x^{4}) + 3 (10 x^{3}) + 3 (- 10 x^{2}) + 3 (5 x) + 3 \cdot - 1$

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

$x^{6} - 5 x^{5} + 10 x^{4} - 10 x^{2 + 1} + x (5 x) + x \cdot - 1 + 3 x^{5} + 3 (- 5 x^{4}) + 3 (10 x^{3}) + 3 (- 10 x^{2}) + 3 (5 x) + 3 \cdot - 1$

Add $2$ and $1$ .

$x^{6} - 5 x^{5} + 10 x^{4} - 10 x^{3} + x (5 x) + x \cdot - 1 + 3 x^{5} + 3 (- 5 x^{4}) + 3 (10 x^{3}) + 3 (- 10 x^{2}) + 3 (5 x) + 3 \cdot - 1$

Rewrite using the commutative property of multiplication.

$x^{6} - 5 x^{5} + 10 x^{4} - 10 x^{3} + 5 x \cdot x + x \cdot - 1 + 3 x^{5} + 3 (- 5 x^{4}) + 3 (10 x^{3}) + 3 (- 10 x^{2}) + 3 (5 x) + 3 \cdot - 1$

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

Move $x$ .

$x^{6} - 5 x^{5} + 10 x^{4} - 10 x^{3} + 5 (x \cdot x) + x \cdot - 1 + 3 x^{5} + 3 (- 5 x^{4}) + 3 (10 x^{3}) + 3 (- 10 x^{2}) + 3 (5 x) + 3 \cdot - 1$

Multiply $x$ by $x$ .

$x^{6} - 5 x^{5} + 10 x^{4} - 10 x^{3} + 5 x^{2} + x \cdot - 1 + 3 x^{5} + 3 (- 5 x^{4}) + 3 (10 x^{3}) + 3 (- 10 x^{2}) + 3 (5 x) + 3 \cdot - 1$

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

$x^{6} - 5 x^{5} + 10 x^{4} - 10 x^{3} + 5 x^{2} - 1 \cdot x + 3 x^{5} + 3 (- 5 x^{4}) + 3 (10 x^{3}) + 3 (- 10 x^{2}) + 3 (5 x) + 3 \cdot - 1$

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

$x^{6} - 5 x^{5} + 10 x^{4} - 10 x^{3} + 5 x^{2} - x + 3 x^{5} + 3 (- 5 x^{4}) + 3 (10 x^{3}) + 3 (- 10 x^{2}) + 3 (5 x) + 3 \cdot - 1$

Multiply $- 5$ by $3$ .

$x^{6} - 5 x^{5} + 10 x^{4} - 10 x^{3} + 5 x^{2} - x + 3 x^{5} - 15 x^{4} + 3 (10 x^{3}) + 3 (- 10 x^{2}) + 3 (5 x) + 3 \cdot - 1$

Multiply $10$ by $3$ .

$x^{6} - 5 x^{5} + 10 x^{4} - 10 x^{3} + 5 x^{2} - x + 3 x^{5} - 15 x^{4} + 30 x^{3} + 3 (- 10 x^{2}) + 3 (5 x) + 3 \cdot - 1$

Multiply $- 10$ by $3$ .

$x^{6} - 5 x^{5} + 10 x^{4} - 10 x^{3} + 5 x^{2} - x + 3 x^{5} - 15 x^{4} + 30 x^{3} - 30 x^{2} + 3 (5 x) + 3 \cdot - 1$

Multiply $5$ by $3$ .

$x^{6} - 5 x^{5} + 10 x^{4} - 10 x^{3} + 5 x^{2} - x + 3 x^{5} - 15 x^{4} + 30 x^{3} - 30 x^{2} + 15 x + 3 \cdot - 1$

Multiply $3$ by $- 1$ .

$x^{6} - 5 x^{5} + 10 x^{4} - 10 x^{3} + 5 x^{2} - x + 3 x^{5} - 15 x^{4} + 30 x^{3} - 30 x^{2} + 15 x - 3$

Simplify by adding terms.

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

$x^{6} - 2 x^{5} + 10 x^{4} - 10 x^{3} + 5 x^{2} - x - 15 x^{4} + 30 x^{3} - 30 x^{2} + 15 x - 3$

Subtract $15 x^{4}$ from $10 x^{4}$ .

$x^{6} - 2 x^{5} - 5 x^{4} - 10 x^{3} + 5 x^{2} - x + 30 x^{3} - 30 x^{2} + 15 x - 3$

Add $- 10 x^{3}$ and $30 x^{3}$ .

$x^{6} - 2 x^{5} - 5 x^{4} + 20 x^{3} + 5 x^{2} - x - 30 x^{2} + 15 x - 3$

Subtract $30 x^{2}$ from $5 x^{2}$ .

$x^{6} - 2 x^{5} - 5 x^{4} + 20 x^{3} - 25 x^{2} - x + 15 x - 3$

Add $- x$ and $15 x$ .

$x^{6} - 2 x^{5} - 5 x^{4} + 20 x^{3} - 25 x^{2} + 14 x - 3$

Step 2

The largest exponent is the degree of the polynomial.

$6$

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

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