Find the End Behavior f(x)=-3x(4x^3-1)(7x^2+1)(x+1)

$f (x) = - 3 x (4 x^{3} - 1) (7 x^{2} + 1) (x + 1)$

Identify the degree of the function.

Simplify and reorder the polynomial.

Simplify terms.

Apply the distributive property.

$(- 3 x (4 x^{3}) - 3 x \cdot - 1) (7 x^{2} + 1) (x + 1)$

Simplify the expression.

Rewrite using the commutative property of multiplication.

$(- 3 \cdot 4 x \cdot x^{3} - 3 x \cdot - 1) (7 x^{2} + 1) (x + 1)$

Multiply $- 1$ by $- 3$ .

$(- 3 \cdot 4 x \cdot x^{3} + 3 x) (7 x^{2} + 1) (x + 1)$

Simplify each term.

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

Move $x^{3}$ .

$(- 3 \cdot 4 (x^{3} x) + 3 x) (7 x^{2} + 1) (x + 1)$

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

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

$(- 3 \cdot 4 (x^{3} x^{1}) + 3 x) (7 x^{2} + 1) (x + 1)$

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

$(- 3 \cdot 4 x^{3 + 1} + 3 x) (7 x^{2} + 1) (x + 1)$

Add $3$ and $1$ .

$(- 3 \cdot 4 x^{4} + 3 x) (7 x^{2} + 1) (x + 1)$

Multiply $- 3$ by $4$ .

$(- 12 x^{4} + 3 x) (7 x^{2} + 1) (x + 1)$

Expand $(- 12 x^{4} + 3 x) (7 x^{2} + 1)$ using the FOIL Method.

Apply the distributive property.

$(- 12 x^{4} (7 x^{2} + 1) + 3 x (7 x^{2} + 1)) (x + 1)$

Apply the distributive property.

$(- 12 x^{4} (7 x^{2}) - 12 x^{4} \cdot 1 + 3 x (7 x^{2} + 1)) (x + 1)$

Apply the distributive property.

$(- 12 x^{4} (7 x^{2}) - 12 x^{4} \cdot 1 + 3 x (7 x^{2}) + 3 x \cdot 1) (x + 1)$

Simplify each term.

Rewrite using the commutative property of multiplication.

$(- 12 \cdot 7 x^{4} x^{2} - 12 x^{4} \cdot 1 + 3 x (7 x^{2}) + 3 x \cdot 1) (x + 1)$

Multiply $x^{4}$ by $x^{2}$ by adding the exponents.

Move $x^{2}$ .

$(- 12 \cdot 7 (x^{2} x^{4}) - 12 x^{4} \cdot 1 + 3 x (7 x^{2}) + 3 x \cdot 1) (x + 1)$

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

$(- 12 \cdot 7 x^{2 + 4} - 12 x^{4} \cdot 1 + 3 x (7 x^{2}) + 3 x \cdot 1) (x + 1)$

Add $2$ and $4$ .

$(- 12 \cdot 7 x^{6} - 12 x^{4} \cdot 1 + 3 x (7 x^{2}) + 3 x \cdot 1) (x + 1)$

Multiply $- 12$ by $7$ .

$(- 84 x^{6} - 12 x^{4} \cdot 1 + 3 x (7 x^{2}) + 3 x \cdot 1) (x + 1)$

Multiply $- 12$ by $1$ .

$(- 84 x^{6} - 12 x^{4} + 3 x (7 x^{2}) + 3 x \cdot 1) (x + 1)$

Rewrite using the commutative property of multiplication.

$(- 84 x^{6} - 12 x^{4} + 3 \cdot 7 x \cdot x^{2} + 3 x \cdot 1) (x + 1)$

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

Move $x^{2}$ .

$(- 84 x^{6} - 12 x^{4} + 3 \cdot 7 (x^{2} x) + 3 x \cdot 1) (x + 1)$

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

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

$(- 84 x^{6} - 12 x^{4} + 3 \cdot 7 (x^{2} x^{1}) + 3 x \cdot 1) (x + 1)$

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

$(- 84 x^{6} - 12 x^{4} + 3 \cdot 7 x^{2 + 1} + 3 x \cdot 1) (x + 1)$

Add $2$ and $1$ .

$(- 84 x^{6} - 12 x^{4} + 3 \cdot 7 x^{3} + 3 x \cdot 1) (x + 1)$

Multiply $3$ by $7$ .

$(- 84 x^{6} - 12 x^{4} + 21 x^{3} + 3 x \cdot 1) (x + 1)$

Multiply $3$ by $1$ .

$(- 84 x^{6} - 12 x^{4} + 21 x^{3} + 3 x) (x + 1)$

Expand $(- 84 x^{6} - 12 x^{4} + 21 x^{3} + 3 x) (x + 1)$ by multiplying each term in the first expression by each term in the second expression.

$- 84 x^{6} x - 84 x^{6} \cdot 1 - 12 x^{4} x - 12 x^{4} \cdot 1 + 21 x^{3} x + 21 x^{3} \cdot 1 + 3 x \cdot x + 3 x \cdot 1$

Simplify terms.

Simplify each term.

Multiply $x^{6}$ by $x$ by adding the exponents.

Move $x$ .

$- 84 (x \cdot x^{6}) - 84 x^{6} \cdot 1 - 12 x^{4} x - 12 x^{4} \cdot 1 + 21 x^{3} x + 21 x^{3} \cdot 1 + 3 x \cdot x + 3 x \cdot 1$

Multiply $x$ by $x^{6}$ .

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

$- 84 (x^{1} x^{6}) - 84 x^{6} \cdot 1 - 12 x^{4} x - 12 x^{4} \cdot 1 + 21 x^{3} x + 21 x^{3} \cdot 1 + 3 x \cdot x + 3 x \cdot 1$

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

$- 84 x^{1 + 6} - 84 x^{6} \cdot 1 - 12 x^{4} x - 12 x^{4} \cdot 1 + 21 x^{3} x + 21 x^{3} \cdot 1 + 3 x \cdot x + 3 x \cdot 1$

Add $1$ and $6$ .

$- 84 x^{7} - 84 x^{6} \cdot 1 - 12 x^{4} x - 12 x^{4} \cdot 1 + 21 x^{3} x + 21 x^{3} \cdot 1 + 3 x \cdot x + 3 x \cdot 1$

Multiply $- 84$ by $1$ .

$- 84 x^{7} - 84 x^{6} - 12 x^{4} x - 12 x^{4} \cdot 1 + 21 x^{3} x + 21 x^{3} \cdot 1 + 3 x \cdot x + 3 x \cdot 1$

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

Move $x$ .

$- 84 x^{7} - 84 x^{6} - 12 (x \cdot x^{4}) - 12 x^{4} \cdot 1 + 21 x^{3} x + 21 x^{3} \cdot 1 + 3 x \cdot x + 3 x \cdot 1$

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

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

$- 84 x^{7} - 84 x^{6} - 12 (x^{1} x^{4}) - 12 x^{4} \cdot 1 + 21 x^{3} x + 21 x^{3} \cdot 1 + 3 x \cdot x + 3 x \cdot 1$

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

$- 84 x^{7} - 84 x^{6} - 12 x^{1 + 4} - 12 x^{4} \cdot 1 + 21 x^{3} x + 21 x^{3} \cdot 1 + 3 x \cdot x + 3 x \cdot 1$

Add $1$ and $4$ .

$- 84 x^{7} - 84 x^{6} - 12 x^{5} - 12 x^{4} \cdot 1 + 21 x^{3} x + 21 x^{3} \cdot 1 + 3 x \cdot x + 3 x \cdot 1$

Multiply $- 12$ by $1$ .

$- 84 x^{7} - 84 x^{6} - 12 x^{5} - 12 x^{4} + 21 x^{3} x + 21 x^{3} \cdot 1 + 3 x \cdot x + 3 x \cdot 1$

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

Move $x$ .

$- 84 x^{7} - 84 x^{6} - 12 x^{5} - 12 x^{4} + 21 (x \cdot x^{3}) + 21 x^{3} \cdot 1 + 3 x \cdot x + 3 x \cdot 1$

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

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

$- 84 x^{7} - 84 x^{6} - 12 x^{5} - 12 x^{4} + 21 (x^{1} x^{3}) + 21 x^{3} \cdot 1 + 3 x \cdot x + 3 x \cdot 1$

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

$- 84 x^{7} - 84 x^{6} - 12 x^{5} - 12 x^{4} + 21 x^{1 + 3} + 21 x^{3} \cdot 1 + 3 x \cdot x + 3 x \cdot 1$

Add $1$ and $3$ .

$- 84 x^{7} - 84 x^{6} - 12 x^{5} - 12 x^{4} + 21 x^{4} + 21 x^{3} \cdot 1 + 3 x \cdot x + 3 x \cdot 1$

Multiply $21$ by $1$ .

$- 84 x^{7} - 84 x^{6} - 12 x^{5} - 12 x^{4} + 21 x^{4} + 21 x^{3} + 3 x \cdot x + 3 x \cdot 1$

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

Move $x$ .

$- 84 x^{7} - 84 x^{6} - 12 x^{5} - 12 x^{4} + 21 x^{4} + 21 x^{3} + 3 (x \cdot x) + 3 x \cdot 1$

Multiply $x$ by $x$ .

$- 84 x^{7} - 84 x^{6} - 12 x^{5} - 12 x^{4} + 21 x^{4} + 21 x^{3} + 3 x^{2} + 3 x \cdot 1$

Multiply $3$ by $1$ .

$- 84 x^{7} - 84 x^{6} - 12 x^{5} - 12 x^{4} + 21 x^{4} + 21 x^{3} + 3 x^{2} + 3 x$

Add $- 12 x^{4}$ and $21 x^{4}$ .

$- 84 x^{7} - 84 x^{6} - 12 x^{5} + 9 x^{4} + 21 x^{3} + 3 x^{2} + 3 x$

The largest exponent is the degree of the polynomial.

$7$

Since the degree is odd, the ends of the function will point in the opposite directions.

Odd

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.

$(- 3 x (4 x^{3}) - 3 x \cdot - 1) (7 x^{2} + 1) (x + 1)$

Simplify the expression.

Rewrite using the commutative property of multiplication.

$(- 3 \cdot 4 x \cdot x^{3} - 3 x \cdot - 1) (7 x^{2} + 1) (x + 1)$

Multiply $- 1$ by $- 3$ .

$(- 3 \cdot 4 x \cdot x^{3} + 3 x) (7 x^{2} + 1) (x + 1)$

Simplify each term.

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

Move $x^{3}$ .

$(- 3 \cdot 4 (x^{3} x) + 3 x) (7 x^{2} + 1) (x + 1)$

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

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

$(- 3 \cdot 4 (x^{3} x^{1}) + 3 x) (7 x^{2} + 1) (x + 1)$

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

$(- 3 \cdot 4 x^{3 + 1} + 3 x) (7 x^{2} + 1) (x + 1)$

Add $3$ and $1$ .

$(- 3 \cdot 4 x^{4} + 3 x) (7 x^{2} + 1) (x + 1)$

Multiply $- 3$ by $4$ .

$(- 12 x^{4} + 3 x) (7 x^{2} + 1) (x + 1)$

Expand $(- 12 x^{4} + 3 x) (7 x^{2} + 1)$ using the FOIL Method.

Apply the distributive property.

$(- 12 x^{4} (7 x^{2} + 1) + 3 x (7 x^{2} + 1)) (x + 1)$

Apply the distributive property.

$(- 12 x^{4} (7 x^{2}) - 12 x^{4} \cdot 1 + 3 x (7 x^{2} + 1)) (x + 1)$

Apply the distributive property.

$(- 12 x^{4} (7 x^{2}) - 12 x^{4} \cdot 1 + 3 x (7 x^{2}) + 3 x \cdot 1) (x + 1)$

Simplify each term.

Rewrite using the commutative property of multiplication.

$(- 12 \cdot 7 x^{4} x^{2} - 12 x^{4} \cdot 1 + 3 x (7 x^{2}) + 3 x \cdot 1) (x + 1)$

Multiply $x^{4}$ by $x^{2}$ by adding the exponents.

Move $x^{2}$ .

$(- 12 \cdot 7 (x^{2} x^{4}) - 12 x^{4} \cdot 1 + 3 x (7 x^{2}) + 3 x \cdot 1) (x + 1)$

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

$(- 12 \cdot 7 x^{2 + 4} - 12 x^{4} \cdot 1 + 3 x (7 x^{2}) + 3 x \cdot 1) (x + 1)$

Add $2$ and $4$ .

$(- 12 \cdot 7 x^{6} - 12 x^{4} \cdot 1 + 3 x (7 x^{2}) + 3 x \cdot 1) (x + 1)$

Multiply $- 12$ by $7$ .

$(- 84 x^{6} - 12 x^{4} \cdot 1 + 3 x (7 x^{2}) + 3 x \cdot 1) (x + 1)$

Multiply $- 12$ by $1$ .

$(- 84 x^{6} - 12 x^{4} + 3 x (7 x^{2}) + 3 x \cdot 1) (x + 1)$

Rewrite using the commutative property of multiplication.

$(- 84 x^{6} - 12 x^{4} + 3 \cdot 7 x \cdot x^{2} + 3 x \cdot 1) (x + 1)$

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

Move $x^{2}$ .

$(- 84 x^{6} - 12 x^{4} + 3 \cdot 7 (x^{2} x) + 3 x \cdot 1) (x + 1)$

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

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

$(- 84 x^{6} - 12 x^{4} + 3 \cdot 7 (x^{2} x^{1}) + 3 x \cdot 1) (x + 1)$

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

$(- 84 x^{6} - 12 x^{4} + 3 \cdot 7 x^{2 + 1} + 3 x \cdot 1) (x + 1)$

Add $2$ and $1$ .

$(- 84 x^{6} - 12 x^{4} + 3 \cdot 7 x^{3} + 3 x \cdot 1) (x + 1)$

Multiply $3$ by $7$ .

$(- 84 x^{6} - 12 x^{4} + 21 x^{3} + 3 x \cdot 1) (x + 1)$

Multiply $3$ by $1$ .

$(- 84 x^{6} - 12 x^{4} + 21 x^{3} + 3 x) (x + 1)$

Expand $(- 84 x^{6} - 12 x^{4} + 21 x^{3} + 3 x) (x + 1)$ by multiplying each term in the first expression by each term in the second expression.

$- 84 x^{6} x - 84 x^{6} \cdot 1 - 12 x^{4} x - 12 x^{4} \cdot 1 + 21 x^{3} x + 21 x^{3} \cdot 1 + 3 x \cdot x + 3 x \cdot 1$

Simplify terms.

Simplify each term.

Multiply $x^{6}$ by $x$ by adding the exponents.

Move $x$ .

$- 84 (x \cdot x^{6}) - 84 x^{6} \cdot 1 - 12 x^{4} x - 12 x^{4} \cdot 1 + 21 x^{3} x + 21 x^{3} \cdot 1 + 3 x \cdot x + 3 x \cdot 1$

Multiply $x$ by $x^{6}$ .

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

$- 84 (x^{1} x^{6}) - 84 x^{6} \cdot 1 - 12 x^{4} x - 12 x^{4} \cdot 1 + 21 x^{3} x + 21 x^{3} \cdot 1 + 3 x \cdot x + 3 x \cdot 1$

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

$- 84 x^{1 + 6} - 84 x^{6} \cdot 1 - 12 x^{4} x - 12 x^{4} \cdot 1 + 21 x^{3} x + 21 x^{3} \cdot 1 + 3 x \cdot x + 3 x \cdot 1$

Add $1$ and $6$ .

$- 84 x^{7} - 84 x^{6} \cdot 1 - 12 x^{4} x - 12 x^{4} \cdot 1 + 21 x^{3} x + 21 x^{3} \cdot 1 + 3 x \cdot x + 3 x \cdot 1$

Multiply $- 84$ by $1$ .

$- 84 x^{7} - 84 x^{6} - 12 x^{4} x - 12 x^{4} \cdot 1 + 21 x^{3} x + 21 x^{3} \cdot 1 + 3 x \cdot x + 3 x \cdot 1$

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

Move $x$ .

$- 84 x^{7} - 84 x^{6} - 12 (x \cdot x^{4}) - 12 x^{4} \cdot 1 + 21 x^{3} x + 21 x^{3} \cdot 1 + 3 x \cdot x + 3 x \cdot 1$

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

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

$- 84 x^{7} - 84 x^{6} - 12 (x^{1} x^{4}) - 12 x^{4} \cdot 1 + 21 x^{3} x + 21 x^{3} \cdot 1 + 3 x \cdot x + 3 x \cdot 1$

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

$- 84 x^{7} - 84 x^{6} - 12 x^{1 + 4} - 12 x^{4} \cdot 1 + 21 x^{3} x + 21 x^{3} \cdot 1 + 3 x \cdot x + 3 x \cdot 1$

Add $1$ and $4$ .

$- 84 x^{7} - 84 x^{6} - 12 x^{5} - 12 x^{4} \cdot 1 + 21 x^{3} x + 21 x^{3} \cdot 1 + 3 x \cdot x + 3 x \cdot 1$

Multiply $- 12$ by $1$ .

$- 84 x^{7} - 84 x^{6} - 12 x^{5} - 12 x^{4} + 21 x^{3} x + 21 x^{3} \cdot 1 + 3 x \cdot x + 3 x \cdot 1$

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

Move $x$ .

$- 84 x^{7} - 84 x^{6} - 12 x^{5} - 12 x^{4} + 21 (x \cdot x^{3}) + 21 x^{3} \cdot 1 + 3 x \cdot x + 3 x \cdot 1$

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

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

$- 84 x^{7} - 84 x^{6} - 12 x^{5} - 12 x^{4} + 21 (x^{1} x^{3}) + 21 x^{3} \cdot 1 + 3 x \cdot x + 3 x \cdot 1$

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

$- 84 x^{7} - 84 x^{6} - 12 x^{5} - 12 x^{4} + 21 x^{1 + 3} + 21 x^{3} \cdot 1 + 3 x \cdot x + 3 x \cdot 1$

Add $1$ and $3$ .

$- 84 x^{7} - 84 x^{6} - 12 x^{5} - 12 x^{4} + 21 x^{4} + 21 x^{3} \cdot 1 + 3 x \cdot x + 3 x \cdot 1$

Multiply $21$ by $1$ .

$- 84 x^{7} - 84 x^{6} - 12 x^{5} - 12 x^{4} + 21 x^{4} + 21 x^{3} + 3 x \cdot x + 3 x \cdot 1$

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

Move $x$ .

$- 84 x^{7} - 84 x^{6} - 12 x^{5} - 12 x^{4} + 21 x^{4} + 21 x^{3} + 3 (x \cdot x) + 3 x \cdot 1$

Multiply $x$ by $x$ .

$- 84 x^{7} - 84 x^{6} - 12 x^{5} - 12 x^{4} + 21 x^{4} + 21 x^{3} + 3 x^{2} + 3 x \cdot 1$

Multiply $3$ by $1$ .

$- 84 x^{7} - 84 x^{6} - 12 x^{5} - 12 x^{4} + 21 x^{4} + 21 x^{3} + 3 x^{2} + 3 x$

Add $- 12 x^{4}$ and $21 x^{4}$ .

$- 84 x^{7} - 84 x^{6} - 12 x^{5} + 9 x^{4} + 21 x^{3} + 3 x^{2} + 3 x$

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

$- 84 x^{7}$

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

$- 84$

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

Negative

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

Determine the behavior.

Rises to the left and falls to the right

Find the End Behavior f(x)=-3x(4x^3-1)(7x^2+1)(x+1)

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