Find the End Behavior f(x)=-2x(3x^2+5)(4x-3)

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

Step 1

Identify the degree of the function.

Simplify and reorder the polynomial.

Simplify terms.

Apply the distributive property.

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

Simplify the expression.

Rewrite using the commutative property of multiplication.

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

Multiply $5$ by $- 2$ .

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

Simplify each term.

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

Move $x^{2}$ .

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

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

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

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

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

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

Add $2$ and $1$ .

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

Multiply $- 2$ by $3$ .

$(- 6 x^{3} - 10 x) (4 x - 3)$

Expand $(- 6 x^{3} - 10 x) (4 x - 3)$ using the FOIL Method.

Apply the distributive property.

$- 6 x^{3} (4 x - 3) - 10 x (4 x - 3)$

Apply the distributive property.

$- 6 x^{3} (4 x) - 6 x^{3} \cdot - 3 - 10 x (4 x - 3)$

Apply the distributive property.

$- 6 x^{3} (4 x) - 6 x^{3} \cdot - 3 - 10 x (4 x) - 10 x \cdot - 3$

Simplify each term.

Rewrite using the commutative property of multiplication.

$- 6 \cdot 4 x^{3} x - 6 x^{3} \cdot - 3 - 10 x (4 x) - 10 x \cdot - 3$

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

Move $x$ .

$- 6 \cdot 4 (x \cdot x^{3}) - 6 x^{3} \cdot - 3 - 10 x (4 x) - 10 x \cdot - 3$

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

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

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

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

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

Add $1$ and $3$ .

$- 6 \cdot 4 x^{4} - 6 x^{3} \cdot - 3 - 10 x (4 x) - 10 x \cdot - 3$

Multiply $- 6$ by $4$ .

$- 24 x^{4} - 6 x^{3} \cdot - 3 - 10 x (4 x) - 10 x \cdot - 3$

Multiply $- 3$ by $- 6$ .

$- 24 x^{4} + 18 x^{3} - 10 x (4 x) - 10 x \cdot - 3$

Rewrite using the commutative property of multiplication.

$- 24 x^{4} + 18 x^{3} - 10 \cdot 4 x \cdot x - 10 x \cdot - 3$

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

Move $x$ .

$- 24 x^{4} + 18 x^{3} - 10 \cdot 4 (x \cdot x) - 10 x \cdot - 3$

Multiply $x$ by $x$ .

$- 24 x^{4} + 18 x^{3} - 10 \cdot 4 x^{2} - 10 x \cdot - 3$

Multiply $- 10$ by $4$ .

$- 24 x^{4} + 18 x^{3} - 40 x^{2} - 10 x \cdot - 3$

Multiply $- 3$ by $- 10$ .

$- 24 x^{4} + 18 x^{3} - 40 x^{2} + 30 x$

The largest exponent is the degree of the polynomial.

$4$

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.

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

Simplify the expression.

Rewrite using the commutative property of multiplication.

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

Multiply $5$ by $- 2$ .

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

Simplify each term.

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

Move $x^{2}$ .

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

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

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

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

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

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

Add $2$ and $1$ .

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

Multiply $- 2$ by $3$ .

$(- 6 x^{3} - 10 x) (4 x - 3)$

Expand $(- 6 x^{3} - 10 x) (4 x - 3)$ using the FOIL Method.

Apply the distributive property.

$- 6 x^{3} (4 x - 3) - 10 x (4 x - 3)$

Apply the distributive property.

$- 6 x^{3} (4 x) - 6 x^{3} \cdot - 3 - 10 x (4 x - 3)$

Apply the distributive property.

$- 6 x^{3} (4 x) - 6 x^{3} \cdot - 3 - 10 x (4 x) - 10 x \cdot - 3$

Simplify each term.

Rewrite using the commutative property of multiplication.

$- 6 \cdot 4 x^{3} x - 6 x^{3} \cdot - 3 - 10 x (4 x) - 10 x \cdot - 3$

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

Move $x$ .

$- 6 \cdot 4 (x \cdot x^{3}) - 6 x^{3} \cdot - 3 - 10 x (4 x) - 10 x \cdot - 3$

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

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

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

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

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

Add $1$ and $3$ .

$- 6 \cdot 4 x^{4} - 6 x^{3} \cdot - 3 - 10 x (4 x) - 10 x \cdot - 3$

Multiply $- 6$ by $4$ .

$- 24 x^{4} - 6 x^{3} \cdot - 3 - 10 x (4 x) - 10 x \cdot - 3$

Multiply $- 3$ by $- 6$ .

$- 24 x^{4} + 18 x^{3} - 10 x (4 x) - 10 x \cdot - 3$

Rewrite using the commutative property of multiplication.

$- 24 x^{4} + 18 x^{3} - 10 \cdot 4 x \cdot x - 10 x \cdot - 3$

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

Move $x$ .

$- 24 x^{4} + 18 x^{3} - 10 \cdot 4 (x \cdot x) - 10 x \cdot - 3$

Multiply $x$ by $x$ .

$- 24 x^{4} + 18 x^{3} - 10 \cdot 4 x^{2} - 10 x \cdot - 3$

Multiply $- 10$ by $4$ .

$- 24 x^{4} + 18 x^{3} - 40 x^{2} - 10 x \cdot - 3$

Multiply $- 3$ by $- 10$ .

$- 24 x^{4} + 18 x^{3} - 40 x^{2} + 30 x$

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

$- 24 x^{4}$

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

$- 24$

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)=-2x(3x^2+5)(4x-3)

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