Find the End Behavior f(x)=-3x^2(3x+5)(2x-1)(x+3)(x-5)

$f (x) = - 3 x^{2} (3 x + 5) (2 x - 1) (x + 3) (x - 5)$

Step 1

Identify the degree of the function.

Simplify and reorder the polynomial.

Simplify terms.

Apply the distributive property.

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

Simplify the expression.

Rewrite using the commutative property of multiplication.

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

Multiply $5$ by $- 3$ .

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

Simplify each term.

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

Move $x$ .

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

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

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

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

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

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

Add $1$ and $2$ .

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

Multiply $- 3$ by $3$ .

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

Expand $(- 9 x^{3} - 15 x^{2}) (2 x - 1)$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

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

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

Move $x$ .

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

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

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

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

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

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

Add $1$ and $3$ .

$(- 9 \cdot 2 x^{4} - 9 x^{3} \cdot - 1 - 15 x^{2} (2 x) - 15 x^{2} \cdot - 1) (x + 3) (x - 5)$

Multiply $- 9$ by $2$ .

$(- 18 x^{4} - 9 x^{3} \cdot - 1 - 15 x^{2} (2 x) - 15 x^{2} \cdot - 1) (x + 3) (x - 5)$

Multiply $- 1$ by $- 9$ .

$(- 18 x^{4} + 9 x^{3} - 15 x^{2} (2 x) - 15 x^{2} \cdot - 1) (x + 3) (x - 5)$

Rewrite using the commutative property of multiplication.

$(- 18 x^{4} + 9 x^{3} - 15 \cdot 2 x^{2} x - 15 x^{2} \cdot - 1) (x + 3) (x - 5)$

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

Move $x$ .

$(- 18 x^{4} + 9 x^{3} - 15 \cdot 2 (x \cdot x^{2}) - 15 x^{2} \cdot - 1) (x + 3) (x - 5)$

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

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

$(- 18 x^{4} + 9 x^{3} - 15 \cdot 2 (x^{1} x^{2}) - 15 x^{2} \cdot - 1) (x + 3) (x - 5)$

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

$(- 18 x^{4} + 9 x^{3} - 15 \cdot 2 x^{1 + 2} - 15 x^{2} \cdot - 1) (x + 3) (x - 5)$

Add $1$ and $2$ .

$(- 18 x^{4} + 9 x^{3} - 15 \cdot 2 x^{3} - 15 x^{2} \cdot - 1) (x + 3) (x - 5)$

Multiply $- 15$ by $2$ .

$(- 18 x^{4} + 9 x^{3} - 30 x^{3} - 15 x^{2} \cdot - 1) (x + 3) (x - 5)$

Multiply $- 1$ by $- 15$ .

$(- 18 x^{4} + 9 x^{3} - 30 x^{3} + 15 x^{2}) (x + 3) (x - 5)$

Subtract $30 x^{3}$ from $9 x^{3}$ .

$(- 18 x^{4} - 21 x^{3} + 15 x^{2}) (x + 3) (x - 5)$

Expand $(- 18 x^{4} - 21 x^{3} + 15 x^{2}) (x + 3)$ by multiplying each term in the first expression by each term in the second expression.

$(- 18 x^{4} x - 18 x^{4} \cdot 3 - 21 x^{3} x - 21 x^{3} \cdot 3 + 15 x^{2} x + 15 x^{2} \cdot 3) (x - 5)$

Simplify terms.

Simplify each term.

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

Move $x$ .

$(- 18 (x \cdot x^{4}) - 18 x^{4} \cdot 3 - 21 x^{3} x - 21 x^{3} \cdot 3 + 15 x^{2} x + 15 x^{2} \cdot 3) (x - 5)$

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

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

$(- 18 (x^{1} x^{4}) - 18 x^{4} \cdot 3 - 21 x^{3} x - 21 x^{3} \cdot 3 + 15 x^{2} x + 15 x^{2} \cdot 3) (x - 5)$

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

$(- 18 x^{1 + 4} - 18 x^{4} \cdot 3 - 21 x^{3} x - 21 x^{3} \cdot 3 + 15 x^{2} x + 15 x^{2} \cdot 3) (x - 5)$

Add $1$ and $4$ .

$(- 18 x^{5} - 18 x^{4} \cdot 3 - 21 x^{3} x - 21 x^{3} \cdot 3 + 15 x^{2} x + 15 x^{2} \cdot 3) (x - 5)$

Multiply $3$ by $- 18$ .

$(- 18 x^{5} - 54 x^{4} - 21 x^{3} x - 21 x^{3} \cdot 3 + 15 x^{2} x + 15 x^{2} \cdot 3) (x - 5)$

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

Move $x$ .

$(- 18 x^{5} - 54 x^{4} - 21 (x \cdot x^{3}) - 21 x^{3} \cdot 3 + 15 x^{2} x + 15 x^{2} \cdot 3) (x - 5)$

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

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

$(- 18 x^{5} - 54 x^{4} - 21 (x^{1} x^{3}) - 21 x^{3} \cdot 3 + 15 x^{2} x + 15 x^{2} \cdot 3) (x - 5)$

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

$(- 18 x^{5} - 54 x^{4} - 21 x^{1 + 3} - 21 x^{3} \cdot 3 + 15 x^{2} x + 15 x^{2} \cdot 3) (x - 5)$

Add $1$ and $3$ .

$(- 18 x^{5} - 54 x^{4} - 21 x^{4} - 21 x^{3} \cdot 3 + 15 x^{2} x + 15 x^{2} \cdot 3) (x - 5)$

Multiply $3$ by $- 21$ .

$(- 18 x^{5} - 54 x^{4} - 21 x^{4} - 63 x^{3} + 15 x^{2} x + 15 x^{2} \cdot 3) (x - 5)$

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

Move $x$ .

$(- 18 x^{5} - 54 x^{4} - 21 x^{4} - 63 x^{3} + 15 (x \cdot x^{2}) + 15 x^{2} \cdot 3) (x - 5)$

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

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

$(- 18 x^{5} - 54 x^{4} - 21 x^{4} - 63 x^{3} + 15 (x^{1} x^{2}) + 15 x^{2} \cdot 3) (x - 5)$

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

$(- 18 x^{5} - 54 x^{4} - 21 x^{4} - 63 x^{3} + 15 x^{1 + 2} + 15 x^{2} \cdot 3) (x - 5)$

Add $1$ and $2$ .

$(- 18 x^{5} - 54 x^{4} - 21 x^{4} - 63 x^{3} + 15 x^{3} + 15 x^{2} \cdot 3) (x - 5)$

Multiply $3$ by $15$ .

$(- 18 x^{5} - 54 x^{4} - 21 x^{4} - 63 x^{3} + 15 x^{3} + 45 x^{2}) (x - 5)$

Simplify by adding terms.

Subtract $21 x^{4}$ from $- 54 x^{4}$ .

$(- 18 x^{5} - 75 x^{4} - 63 x^{3} + 15 x^{3} + 45 x^{2}) (x - 5)$

Add $- 63 x^{3}$ and $15 x^{3}$ .

$(- 18 x^{5} - 75 x^{4} - 48 x^{3} + 45 x^{2}) (x - 5)$

Expand $(- 18 x^{5} - 75 x^{4} - 48 x^{3} + 45 x^{2}) (x - 5)$ by multiplying each term in the first expression by each term in the second expression.

$- 18 x^{5} x - 18 x^{5} \cdot - 5 - 75 x^{4} x - 75 x^{4} \cdot - 5 - 48 x^{3} x - 48 x^{3} \cdot - 5 + 45 x^{2} x + 45 x^{2} \cdot - 5$

Simplify terms.

Simplify each term.

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

Move $x$ .

$- 18 (x \cdot x^{5}) - 18 x^{5} \cdot - 5 - 75 x^{4} x - 75 x^{4} \cdot - 5 - 48 x^{3} x - 48 x^{3} \cdot - 5 + 45 x^{2} x + 45 x^{2} \cdot - 5$

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

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

$- 18 (x^{1} x^{5}) - 18 x^{5} \cdot - 5 - 75 x^{4} x - 75 x^{4} \cdot - 5 - 48 x^{3} x - 48 x^{3} \cdot - 5 + 45 x^{2} x + 45 x^{2} \cdot - 5$

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

$- 18 x^{1 + 5} - 18 x^{5} \cdot - 5 - 75 x^{4} x - 75 x^{4} \cdot - 5 - 48 x^{3} x - 48 x^{3} \cdot - 5 + 45 x^{2} x + 45 x^{2} \cdot - 5$

Add $1$ and $5$ .

$- 18 x^{6} - 18 x^{5} \cdot - 5 - 75 x^{4} x - 75 x^{4} \cdot - 5 - 48 x^{3} x - 48 x^{3} \cdot - 5 + 45 x^{2} x + 45 x^{2} \cdot - 5$

Multiply $- 5$ by $- 18$ .

$- 18 x^{6} + 90 x^{5} - 75 x^{4} x - 75 x^{4} \cdot - 5 - 48 x^{3} x - 48 x^{3} \cdot - 5 + 45 x^{2} x + 45 x^{2} \cdot - 5$

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

Move $x$ .

$- 18 x^{6} + 90 x^{5} - 75 (x \cdot x^{4}) - 75 x^{4} \cdot - 5 - 48 x^{3} x - 48 x^{3} \cdot - 5 + 45 x^{2} x + 45 x^{2} \cdot - 5$

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

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

$- 18 x^{6} + 90 x^{5} - 75 (x^{1} x^{4}) - 75 x^{4} \cdot - 5 - 48 x^{3} x - 48 x^{3} \cdot - 5 + 45 x^{2} x + 45 x^{2} \cdot - 5$

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

$- 18 x^{6} + 90 x^{5} - 75 x^{1 + 4} - 75 x^{4} \cdot - 5 - 48 x^{3} x - 48 x^{3} \cdot - 5 + 45 x^{2} x + 45 x^{2} \cdot - 5$

Add $1$ and $4$ .

$- 18 x^{6} + 90 x^{5} - 75 x^{5} - 75 x^{4} \cdot - 5 - 48 x^{3} x - 48 x^{3} \cdot - 5 + 45 x^{2} x + 45 x^{2} \cdot - 5$

Multiply $- 5$ by $- 75$ .

$- 18 x^{6} + 90 x^{5} - 75 x^{5} + 375 x^{4} - 48 x^{3} x - 48 x^{3} \cdot - 5 + 45 x^{2} x + 45 x^{2} \cdot - 5$

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

Move $x$ .

$- 18 x^{6} + 90 x^{5} - 75 x^{5} + 375 x^{4} - 48 (x \cdot x^{3}) - 48 x^{3} \cdot - 5 + 45 x^{2} x + 45 x^{2} \cdot - 5$

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

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

$- 18 x^{6} + 90 x^{5} - 75 x^{5} + 375 x^{4} - 48 (x^{1} x^{3}) - 48 x^{3} \cdot - 5 + 45 x^{2} x + 45 x^{2} \cdot - 5$

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

$- 18 x^{6} + 90 x^{5} - 75 x^{5} + 375 x^{4} - 48 x^{1 + 3} - 48 x^{3} \cdot - 5 + 45 x^{2} x + 45 x^{2} \cdot - 5$

Add $1$ and $3$ .

$- 18 x^{6} + 90 x^{5} - 75 x^{5} + 375 x^{4} - 48 x^{4} - 48 x^{3} \cdot - 5 + 45 x^{2} x + 45 x^{2} \cdot - 5$

Multiply $- 5$ by $- 48$ .

$- 18 x^{6} + 90 x^{5} - 75 x^{5} + 375 x^{4} - 48 x^{4} + 240 x^{3} + 45 x^{2} x + 45 x^{2} \cdot - 5$

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

Move $x$ .

$- 18 x^{6} + 90 x^{5} - 75 x^{5} + 375 x^{4} - 48 x^{4} + 240 x^{3} + 45 (x \cdot x^{2}) + 45 x^{2} \cdot - 5$

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

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

$- 18 x^{6} + 90 x^{5} - 75 x^{5} + 375 x^{4} - 48 x^{4} + 240 x^{3} + 45 (x^{1} x^{2}) + 45 x^{2} \cdot - 5$

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

$- 18 x^{6} + 90 x^{5} - 75 x^{5} + 375 x^{4} - 48 x^{4} + 240 x^{3} + 45 x^{1 + 2} + 45 x^{2} \cdot - 5$

Add $1$ and $2$ .

$- 18 x^{6} + 90 x^{5} - 75 x^{5} + 375 x^{4} - 48 x^{4} + 240 x^{3} + 45 x^{3} + 45 x^{2} \cdot - 5$

Multiply $- 5$ by $45$ .

$- 18 x^{6} + 90 x^{5} - 75 x^{5} + 375 x^{4} - 48 x^{4} + 240 x^{3} + 45 x^{3} - 225 x^{2}$

Simplify by adding terms.

Subtract $75 x^{5}$ from $90 x^{5}$ .

$- 18 x^{6} + 15 x^{5} + 375 x^{4} - 48 x^{4} + 240 x^{3} + 45 x^{3} - 225 x^{2}$

Subtract $48 x^{4}$ from $375 x^{4}$ .

$- 18 x^{6} + 15 x^{5} + 327 x^{4} + 240 x^{3} + 45 x^{3} - 225 x^{2}$

Add $240 x^{3}$ and $45 x^{3}$ .

$- 18 x^{6} + 15 x^{5} + 327 x^{4} + 285 x^{3} - 225 x^{2}$

The largest exponent is the degree of the polynomial.

$6$

Step 2

Since the degree is even, the ends of the function will point in the same direction.

Even

Step 3

Identify the leading coefficient.

Simplify the polynomial, then reorder it left to right starting with the highest degree term.

Simplify terms.

Apply the distributive property.

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

Simplify the expression.

Rewrite using the commutative property of multiplication.

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

Multiply $5$ by $- 3$ .

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

Simplify each term.

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

Move $x$ .

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

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

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

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

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

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

Add $1$ and $2$ .

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

Multiply $- 3$ by $3$ .

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

Expand $(- 9 x^{3} - 15 x^{2}) (2 x - 1)$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

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

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

Move $x$ .

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

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

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

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

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

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

Add $1$ and $3$ .

$(- 9 \cdot 2 x^{4} - 9 x^{3} \cdot - 1 - 15 x^{2} (2 x) - 15 x^{2} \cdot - 1) (x + 3) (x - 5)$

Multiply $- 9$ by $2$ .

$(- 18 x^{4} - 9 x^{3} \cdot - 1 - 15 x^{2} (2 x) - 15 x^{2} \cdot - 1) (x + 3) (x - 5)$

Multiply $- 1$ by $- 9$ .

$(- 18 x^{4} + 9 x^{3} - 15 x^{2} (2 x) - 15 x^{2} \cdot - 1) (x + 3) (x - 5)$

Rewrite using the commutative property of multiplication.

$(- 18 x^{4} + 9 x^{3} - 15 \cdot 2 x^{2} x - 15 x^{2} \cdot - 1) (x + 3) (x - 5)$

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

Move $x$ .

$(- 18 x^{4} + 9 x^{3} - 15 \cdot 2 (x \cdot x^{2}) - 15 x^{2} \cdot - 1) (x + 3) (x - 5)$

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

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

$(- 18 x^{4} + 9 x^{3} - 15 \cdot 2 (x^{1} x^{2}) - 15 x^{2} \cdot - 1) (x + 3) (x - 5)$

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

$(- 18 x^{4} + 9 x^{3} - 15 \cdot 2 x^{1 + 2} - 15 x^{2} \cdot - 1) (x + 3) (x - 5)$

Add $1$ and $2$ .

$(- 18 x^{4} + 9 x^{3} - 15 \cdot 2 x^{3} - 15 x^{2} \cdot - 1) (x + 3) (x - 5)$

Multiply $- 15$ by $2$ .

$(- 18 x^{4} + 9 x^{3} - 30 x^{3} - 15 x^{2} \cdot - 1) (x + 3) (x - 5)$

Multiply $- 1$ by $- 15$ .

$(- 18 x^{4} + 9 x^{3} - 30 x^{3} + 15 x^{2}) (x + 3) (x - 5)$

Subtract $30 x^{3}$ from $9 x^{3}$ .

$(- 18 x^{4} - 21 x^{3} + 15 x^{2}) (x + 3) (x - 5)$

Expand $(- 18 x^{4} - 21 x^{3} + 15 x^{2}) (x + 3)$ by multiplying each term in the first expression by each term in the second expression.

$(- 18 x^{4} x - 18 x^{4} \cdot 3 - 21 x^{3} x - 21 x^{3} \cdot 3 + 15 x^{2} x + 15 x^{2} \cdot 3) (x - 5)$

Simplify terms.

Simplify each term.

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

Move $x$ .

$(- 18 (x \cdot x^{4}) - 18 x^{4} \cdot 3 - 21 x^{3} x - 21 x^{3} \cdot 3 + 15 x^{2} x + 15 x^{2} \cdot 3) (x - 5)$

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

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

$(- 18 (x^{1} x^{4}) - 18 x^{4} \cdot 3 - 21 x^{3} x - 21 x^{3} \cdot 3 + 15 x^{2} x + 15 x^{2} \cdot 3) (x - 5)$

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

$(- 18 x^{1 + 4} - 18 x^{4} \cdot 3 - 21 x^{3} x - 21 x^{3} \cdot 3 + 15 x^{2} x + 15 x^{2} \cdot 3) (x - 5)$

Add $1$ and $4$ .

$(- 18 x^{5} - 18 x^{4} \cdot 3 - 21 x^{3} x - 21 x^{3} \cdot 3 + 15 x^{2} x + 15 x^{2} \cdot 3) (x - 5)$

Multiply $3$ by $- 18$ .

$(- 18 x^{5} - 54 x^{4} - 21 x^{3} x - 21 x^{3} \cdot 3 + 15 x^{2} x + 15 x^{2} \cdot 3) (x - 5)$

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

Move $x$ .

$(- 18 x^{5} - 54 x^{4} - 21 (x \cdot x^{3}) - 21 x^{3} \cdot 3 + 15 x^{2} x + 15 x^{2} \cdot 3) (x - 5)$

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

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

$(- 18 x^{5} - 54 x^{4} - 21 (x^{1} x^{3}) - 21 x^{3} \cdot 3 + 15 x^{2} x + 15 x^{2} \cdot 3) (x - 5)$

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

$(- 18 x^{5} - 54 x^{4} - 21 x^{1 + 3} - 21 x^{3} \cdot 3 + 15 x^{2} x + 15 x^{2} \cdot 3) (x - 5)$

Add $1$ and $3$ .

$(- 18 x^{5} - 54 x^{4} - 21 x^{4} - 21 x^{3} \cdot 3 + 15 x^{2} x + 15 x^{2} \cdot 3) (x - 5)$

Multiply $3$ by $- 21$ .

$(- 18 x^{5} - 54 x^{4} - 21 x^{4} - 63 x^{3} + 15 x^{2} x + 15 x^{2} \cdot 3) (x - 5)$

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

Move $x$ .

$(- 18 x^{5} - 54 x^{4} - 21 x^{4} - 63 x^{3} + 15 (x \cdot x^{2}) + 15 x^{2} \cdot 3) (x - 5)$

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

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

$(- 18 x^{5} - 54 x^{4} - 21 x^{4} - 63 x^{3} + 15 (x^{1} x^{2}) + 15 x^{2} \cdot 3) (x - 5)$

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

$(- 18 x^{5} - 54 x^{4} - 21 x^{4} - 63 x^{3} + 15 x^{1 + 2} + 15 x^{2} \cdot 3) (x - 5)$

Add $1$ and $2$ .

$(- 18 x^{5} - 54 x^{4} - 21 x^{4} - 63 x^{3} + 15 x^{3} + 15 x^{2} \cdot 3) (x - 5)$

Multiply $3$ by $15$ .

$(- 18 x^{5} - 54 x^{4} - 21 x^{4} - 63 x^{3} + 15 x^{3} + 45 x^{2}) (x - 5)$

Simplify by adding terms.

Subtract $21 x^{4}$ from $- 54 x^{4}$ .

$(- 18 x^{5} - 75 x^{4} - 63 x^{3} + 15 x^{3} + 45 x^{2}) (x - 5)$

Add $- 63 x^{3}$ and $15 x^{3}$ .

$(- 18 x^{5} - 75 x^{4} - 48 x^{3} + 45 x^{2}) (x - 5)$

Expand $(- 18 x^{5} - 75 x^{4} - 48 x^{3} + 45 x^{2}) (x - 5)$ by multiplying each term in the first expression by each term in the second expression.

$- 18 x^{5} x - 18 x^{5} \cdot - 5 - 75 x^{4} x - 75 x^{4} \cdot - 5 - 48 x^{3} x - 48 x^{3} \cdot - 5 + 45 x^{2} x + 45 x^{2} \cdot - 5$

Simplify terms.

Simplify each term.

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

Move $x$ .

$- 18 (x \cdot x^{5}) - 18 x^{5} \cdot - 5 - 75 x^{4} x - 75 x^{4} \cdot - 5 - 48 x^{3} x - 48 x^{3} \cdot - 5 + 45 x^{2} x + 45 x^{2} \cdot - 5$

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

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

$- 18 (x^{1} x^{5}) - 18 x^{5} \cdot - 5 - 75 x^{4} x - 75 x^{4} \cdot - 5 - 48 x^{3} x - 48 x^{3} \cdot - 5 + 45 x^{2} x + 45 x^{2} \cdot - 5$

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

$- 18 x^{1 + 5} - 18 x^{5} \cdot - 5 - 75 x^{4} x - 75 x^{4} \cdot - 5 - 48 x^{3} x - 48 x^{3} \cdot - 5 + 45 x^{2} x + 45 x^{2} \cdot - 5$

Add $1$ and $5$ .

$- 18 x^{6} - 18 x^{5} \cdot - 5 - 75 x^{4} x - 75 x^{4} \cdot - 5 - 48 x^{3} x - 48 x^{3} \cdot - 5 + 45 x^{2} x + 45 x^{2} \cdot - 5$

Multiply $- 5$ by $- 18$ .

$- 18 x^{6} + 90 x^{5} - 75 x^{4} x - 75 x^{4} \cdot - 5 - 48 x^{3} x - 48 x^{3} \cdot - 5 + 45 x^{2} x + 45 x^{2} \cdot - 5$

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

Move $x$ .

$- 18 x^{6} + 90 x^{5} - 75 (x \cdot x^{4}) - 75 x^{4} \cdot - 5 - 48 x^{3} x - 48 x^{3} \cdot - 5 + 45 x^{2} x + 45 x^{2} \cdot - 5$

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

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

$- 18 x^{6} + 90 x^{5} - 75 (x^{1} x^{4}) - 75 x^{4} \cdot - 5 - 48 x^{3} x - 48 x^{3} \cdot - 5 + 45 x^{2} x + 45 x^{2} \cdot - 5$

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

$- 18 x^{6} + 90 x^{5} - 75 x^{1 + 4} - 75 x^{4} \cdot - 5 - 48 x^{3} x - 48 x^{3} \cdot - 5 + 45 x^{2} x + 45 x^{2} \cdot - 5$

Add $1$ and $4$ .

$- 18 x^{6} + 90 x^{5} - 75 x^{5} - 75 x^{4} \cdot - 5 - 48 x^{3} x - 48 x^{3} \cdot - 5 + 45 x^{2} x + 45 x^{2} \cdot - 5$

Multiply $- 5$ by $- 75$ .

$- 18 x^{6} + 90 x^{5} - 75 x^{5} + 375 x^{4} - 48 x^{3} x - 48 x^{3} \cdot - 5 + 45 x^{2} x + 45 x^{2} \cdot - 5$

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

Move $x$ .

$- 18 x^{6} + 90 x^{5} - 75 x^{5} + 375 x^{4} - 48 (x \cdot x^{3}) - 48 x^{3} \cdot - 5 + 45 x^{2} x + 45 x^{2} \cdot - 5$

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

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

$- 18 x^{6} + 90 x^{5} - 75 x^{5} + 375 x^{4} - 48 (x^{1} x^{3}) - 48 x^{3} \cdot - 5 + 45 x^{2} x + 45 x^{2} \cdot - 5$

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

$- 18 x^{6} + 90 x^{5} - 75 x^{5} + 375 x^{4} - 48 x^{1 + 3} - 48 x^{3} \cdot - 5 + 45 x^{2} x + 45 x^{2} \cdot - 5$

Add $1$ and $3$ .

$- 18 x^{6} + 90 x^{5} - 75 x^{5} + 375 x^{4} - 48 x^{4} - 48 x^{3} \cdot - 5 + 45 x^{2} x + 45 x^{2} \cdot - 5$

Multiply $- 5$ by $- 48$ .

$- 18 x^{6} + 90 x^{5} - 75 x^{5} + 375 x^{4} - 48 x^{4} + 240 x^{3} + 45 x^{2} x + 45 x^{2} \cdot - 5$

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

Move $x$ .

$- 18 x^{6} + 90 x^{5} - 75 x^{5} + 375 x^{4} - 48 x^{4} + 240 x^{3} + 45 (x \cdot x^{2}) + 45 x^{2} \cdot - 5$

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

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

$- 18 x^{6} + 90 x^{5} - 75 x^{5} + 375 x^{4} - 48 x^{4} + 240 x^{3} + 45 (x^{1} x^{2}) + 45 x^{2} \cdot - 5$

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

$- 18 x^{6} + 90 x^{5} - 75 x^{5} + 375 x^{4} - 48 x^{4} + 240 x^{3} + 45 x^{1 + 2} + 45 x^{2} \cdot - 5$

Add $1$ and $2$ .

$- 18 x^{6} + 90 x^{5} - 75 x^{5} + 375 x^{4} - 48 x^{4} + 240 x^{3} + 45 x^{3} + 45 x^{2} \cdot - 5$

Multiply $- 5$ by $45$ .

$- 18 x^{6} + 90 x^{5} - 75 x^{5} + 375 x^{4} - 48 x^{4} + 240 x^{3} + 45 x^{3} - 225 x^{2}$

Simplify by adding terms.

Subtract $75 x^{5}$ from $90 x^{5}$ .

$- 18 x^{6} + 15 x^{5} + 375 x^{4} - 48 x^{4} + 240 x^{3} + 45 x^{3} - 225 x^{2}$

Subtract $48 x^{4}$ from $375 x^{4}$ .

$- 18 x^{6} + 15 x^{5} + 327 x^{4} + 240 x^{3} + 45 x^{3} - 225 x^{2}$

Add $240 x^{3}$ and $45 x^{3}$ .

$- 18 x^{6} + 15 x^{5} + 327 x^{4} + 285 x^{3} - 225 x^{2}$

The leading term in a polynomial is the term with the highest degree.

$- 18 x^{6}$

The leading coefficient in a polynomial is the coefficient of the leading term.

$- 18$

Step 4

Since the leading coefficient is negative, the graph falls to the right.

Negative

Step 5

Use the degree of the function, as well as the sign of the leading coefficient to determine the behavior.

1. Even and Positive: Rises to the left and rises to the right.

2. Even and Negative: Falls to the left and falls to the right.

3. Odd and Positive: Falls to the left and rises to the right.

4. Odd and Negative: Rises to the left and falls to the right

Step 6

Determine the behavior.

Falls to the left and falls to the right

Step 7

Find the End Behavior f(x)=-3x^2(3x+5)(2x-1)(x+3)(x-5)

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