Find the Degree f(x)=(x+3)(x-2)^5

$f (x) = (x + 3) {(x - 2)}^{5}$

Step 1

Simplify and reorder the polynomial.

Use the Binomial Theorem.

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

Simplify each term.

Multiply $- 2$ by $5$ .

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

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

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

Multiply $4$ by $10$ .

$(x + 3) (x^{5} - 10 x^{4} + 40 x^{3} + 10 x^{2} {(- 2)}^{3} + 5 x {(- 2)}^{4} + {(- 2)}^{5})$

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

$(x + 3) (x^{5} - 10 x^{4} + 40 x^{3} + 10 x^{2} \cdot - 8 + 5 x {(- 2)}^{4} + {(- 2)}^{5})$

Multiply $- 8$ by $10$ .

$(x + 3) (x^{5} - 10 x^{4} + 40 x^{3} - 80 x^{2} + 5 x {(- 2)}^{4} + {(- 2)}^{5})$

Raise $- 2$ to the power of $4$ .

$(x + 3) (x^{5} - 10 x^{4} + 40 x^{3} - 80 x^{2} + 5 x \cdot 16 + {(- 2)}^{5})$

Multiply $16$ by $5$ .

$(x + 3) (x^{5} - 10 x^{4} + 40 x^{3} - 80 x^{2} + 80 x + {(- 2)}^{5})$

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

$(x + 3) (x^{5} - 10 x^{4} + 40 x^{3} - 80 x^{2} + 80 x - 32)$

Expand $(x + 3) (x^{5} - 10 x^{4} + 40 x^{3} - 80 x^{2} + 80 x - 32)$ by multiplying each term in the first expression by each term in the second expression.

$x \cdot x^{5} + x (- 10 x^{4}) + x (40 x^{3}) + x (- 80 x^{2}) + x (80 x) + x \cdot - 32 + 3 x^{5} + 3 (- 10 x^{4}) + 3 (40 x^{3}) + 3 (- 80 x^{2}) + 3 (80 x) + 3 \cdot - 32$

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 $1$ .

$x^{1} x^{5} + x (- 10 x^{4}) + x (40 x^{3}) + x (- 80 x^{2}) + x (80 x) + x \cdot - 32 + 3 x^{5} + 3 (- 10 x^{4}) + 3 (40 x^{3}) + 3 (- 80 x^{2}) + 3 (80 x) + 3 \cdot - 32$

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

$x^{1 + 5} + x (- 10 x^{4}) + x (40 x^{3}) + x (- 80 x^{2}) + x (80 x) + x \cdot - 32 + 3 x^{5} + 3 (- 10 x^{4}) + 3 (40 x^{3}) + 3 (- 80 x^{2}) + 3 (80 x) + 3 \cdot - 32$

Add $1$ and $5$ .

$x^{6} + x (- 10 x^{4}) + x (40 x^{3}) + x (- 80 x^{2}) + x (80 x) + x \cdot - 32 + 3 x^{5} + 3 (- 10 x^{4}) + 3 (40 x^{3}) + 3 (- 80 x^{2}) + 3 (80 x) + 3 \cdot - 32$

Rewrite using the commutative property of multiplication.

$x^{6} - 10 x \cdot x^{4} + x (40 x^{3}) + x (- 80 x^{2}) + x (80 x) + x \cdot - 32 + 3 x^{5} + 3 (- 10 x^{4}) + 3 (40 x^{3}) + 3 (- 80 x^{2}) + 3 (80 x) + 3 \cdot - 32$

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

Move $x^{4}$ .

$x^{6} - 10 (x^{4} x) + x (40 x^{3}) + x (- 80 x^{2}) + x (80 x) + x \cdot - 32 + 3 x^{5} + 3 (- 10 x^{4}) + 3 (40 x^{3}) + 3 (- 80 x^{2}) + 3 (80 x) + 3 \cdot - 32$

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

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

$x^{6} - 10 (x^{4} x^{1}) + x (40 x^{3}) + x (- 80 x^{2}) + x (80 x) + x \cdot - 32 + 3 x^{5} + 3 (- 10 x^{4}) + 3 (40 x^{3}) + 3 (- 80 x^{2}) + 3 (80 x) + 3 \cdot - 32$

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

$x^{6} - 10 x^{4 + 1} + x (40 x^{3}) + x (- 80 x^{2}) + x (80 x) + x \cdot - 32 + 3 x^{5} + 3 (- 10 x^{4}) + 3 (40 x^{3}) + 3 (- 80 x^{2}) + 3 (80 x) + 3 \cdot - 32$

Add $4$ and $1$ .

$x^{6} - 10 x^{5} + x (40 x^{3}) + x (- 80 x^{2}) + x (80 x) + x \cdot - 32 + 3 x^{5} + 3 (- 10 x^{4}) + 3 (40 x^{3}) + 3 (- 80 x^{2}) + 3 (80 x) + 3 \cdot - 32$

Rewrite using the commutative property of multiplication.

$x^{6} - 10 x^{5} + 40 x \cdot x^{3} + x (- 80 x^{2}) + x (80 x) + x \cdot - 32 + 3 x^{5} + 3 (- 10 x^{4}) + 3 (40 x^{3}) + 3 (- 80 x^{2}) + 3 (80 x) + 3 \cdot - 32$

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

Move $x^{3}$ .

$x^{6} - 10 x^{5} + 40 (x^{3} x) + x (- 80 x^{2}) + x (80 x) + x \cdot - 32 + 3 x^{5} + 3 (- 10 x^{4}) + 3 (40 x^{3}) + 3 (- 80 x^{2}) + 3 (80 x) + 3 \cdot - 32$

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

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

$x^{6} - 10 x^{5} + 40 (x^{3} x^{1}) + x (- 80 x^{2}) + x (80 x) + x \cdot - 32 + 3 x^{5} + 3 (- 10 x^{4}) + 3 (40 x^{3}) + 3 (- 80 x^{2}) + 3 (80 x) + 3 \cdot - 32$

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

$x^{6} - 10 x^{5} + 40 x^{3 + 1} + x (- 80 x^{2}) + x (80 x) + x \cdot - 32 + 3 x^{5} + 3 (- 10 x^{4}) + 3 (40 x^{3}) + 3 (- 80 x^{2}) + 3 (80 x) + 3 \cdot - 32$

Add $3$ and $1$ .

$x^{6} - 10 x^{5} + 40 x^{4} + x (- 80 x^{2}) + x (80 x) + x \cdot - 32 + 3 x^{5} + 3 (- 10 x^{4}) + 3 (40 x^{3}) + 3 (- 80 x^{2}) + 3 (80 x) + 3 \cdot - 32$

Rewrite using the commutative property of multiplication.

$x^{6} - 10 x^{5} + 40 x^{4} - 80 x \cdot x^{2} + x (80 x) + x \cdot - 32 + 3 x^{5} + 3 (- 10 x^{4}) + 3 (40 x^{3}) + 3 (- 80 x^{2}) + 3 (80 x) + 3 \cdot - 32$

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

Move $x^{2}$ .

$x^{6} - 10 x^{5} + 40 x^{4} - 80 (x^{2} x) + x (80 x) + x \cdot - 32 + 3 x^{5} + 3 (- 10 x^{4}) + 3 (40 x^{3}) + 3 (- 80 x^{2}) + 3 (80 x) + 3 \cdot - 32$

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

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

$x^{6} - 10 x^{5} + 40 x^{4} - 80 (x^{2} x^{1}) + x (80 x) + x \cdot - 32 + 3 x^{5} + 3 (- 10 x^{4}) + 3 (40 x^{3}) + 3 (- 80 x^{2}) + 3 (80 x) + 3 \cdot - 32$

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

$x^{6} - 10 x^{5} + 40 x^{4} - 80 x^{2 + 1} + x (80 x) + x \cdot - 32 + 3 x^{5} + 3 (- 10 x^{4}) + 3 (40 x^{3}) + 3 (- 80 x^{2}) + 3 (80 x) + 3 \cdot - 32$

Add $2$ and $1$ .

$x^{6} - 10 x^{5} + 40 x^{4} - 80 x^{3} + x (80 x) + x \cdot - 32 + 3 x^{5} + 3 (- 10 x^{4}) + 3 (40 x^{3}) + 3 (- 80 x^{2}) + 3 (80 x) + 3 \cdot - 32$

Rewrite using the commutative property of multiplication.

$x^{6} - 10 x^{5} + 40 x^{4} - 80 x^{3} + 80 x \cdot x + x \cdot - 32 + 3 x^{5} + 3 (- 10 x^{4}) + 3 (40 x^{3}) + 3 (- 80 x^{2}) + 3 (80 x) + 3 \cdot - 32$

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

Move $x$ .

$x^{6} - 10 x^{5} + 40 x^{4} - 80 x^{3} + 80 (x \cdot x) + x \cdot - 32 + 3 x^{5} + 3 (- 10 x^{4}) + 3 (40 x^{3}) + 3 (- 80 x^{2}) + 3 (80 x) + 3 \cdot - 32$

Multiply $x$ by $x$ .

$x^{6} - 10 x^{5} + 40 x^{4} - 80 x^{3} + 80 x^{2} + x \cdot - 32 + 3 x^{5} + 3 (- 10 x^{4}) + 3 (40 x^{3}) + 3 (- 80 x^{2}) + 3 (80 x) + 3 \cdot - 32$

Move $- 32$ to the left of $x$ .

$x^{6} - 10 x^{5} + 40 x^{4} - 80 x^{3} + 80 x^{2} - 32 \cdot x + 3 x^{5} + 3 (- 10 x^{4}) + 3 (40 x^{3}) + 3 (- 80 x^{2}) + 3 (80 x) + 3 \cdot - 32$

Multiply $- 10$ by $3$ .

$x^{6} - 10 x^{5} + 40 x^{4} - 80 x^{3} + 80 x^{2} - 32 x + 3 x^{5} - 30 x^{4} + 3 (40 x^{3}) + 3 (- 80 x^{2}) + 3 (80 x) + 3 \cdot - 32$

Multiply $40$ by $3$ .

$x^{6} - 10 x^{5} + 40 x^{4} - 80 x^{3} + 80 x^{2} - 32 x + 3 x^{5} - 30 x^{4} + 120 x^{3} + 3 (- 80 x^{2}) + 3 (80 x) + 3 \cdot - 32$

Multiply $- 80$ by $3$ .

$x^{6} - 10 x^{5} + 40 x^{4} - 80 x^{3} + 80 x^{2} - 32 x + 3 x^{5} - 30 x^{4} + 120 x^{3} - 240 x^{2} + 3 (80 x) + 3 \cdot - 32$

Multiply $80$ by $3$ .

$x^{6} - 10 x^{5} + 40 x^{4} - 80 x^{3} + 80 x^{2} - 32 x + 3 x^{5} - 30 x^{4} + 120 x^{3} - 240 x^{2} + 240 x + 3 \cdot - 32$

Multiply $3$ by $- 32$ .

$x^{6} - 10 x^{5} + 40 x^{4} - 80 x^{3} + 80 x^{2} - 32 x + 3 x^{5} - 30 x^{4} + 120 x^{3} - 240 x^{2} + 240 x - 96$

Simplify by adding terms.

Add $- 10 x^{5}$ and $3 x^{5}$ .

$x^{6} - 7 x^{5} + 40 x^{4} - 80 x^{3} + 80 x^{2} - 32 x - 30 x^{4} + 120 x^{3} - 240 x^{2} + 240 x - 96$

Subtract $30 x^{4}$ from $40 x^{4}$ .

$x^{6} - 7 x^{5} + 10 x^{4} - 80 x^{3} + 80 x^{2} - 32 x + 120 x^{3} - 240 x^{2} + 240 x - 96$

Add $- 80 x^{3}$ and $120 x^{3}$ .

$x^{6} - 7 x^{5} + 10 x^{4} + 40 x^{3} + 80 x^{2} - 32 x - 240 x^{2} + 240 x - 96$

Subtract $240 x^{2}$ from $80 x^{2}$ .

$x^{6} - 7 x^{5} + 10 x^{4} + 40 x^{3} - 160 x^{2} - 32 x + 240 x - 96$

Add $- 32 x$ and $240 x$ .

$x^{6} - 7 x^{5} + 10 x^{4} + 40 x^{3} - 160 x^{2} + 208 x - 96$

Step 2

The largest exponent is the degree of the polynomial.

$6$

Find the Degree f(x)=(x+3)(x-2)^5

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