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

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

Step 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.

$7$

Step 2

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

Odd

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.

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

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.

$- 84$

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.

Rises to the left and falls to the right

Step 7

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

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