Find the End Behavior f(x)=-1.9(x+5)(x-7)(2x+1)

$f (x) = - 1.9 (x + 5) (x - 7) (2 x + 1)$

Step 1

Identify the degree of the function.

Simplify and reorder the polynomial.

Simplify by multiplying through.

Apply the distributive property.

$(- 1.9 x - 1.9 \cdot 5) (x - 7) (2 x + 1)$

Multiply $- 1.9$ by $5$ .

$(- 1.9 x - 9.5) (x - 7) (2 x + 1)$

Expand $(- 1.9 x - 9.5) (x - 7)$ using the FOIL Method.

Apply the distributive property.

$(- 1.9 x (x - 7) - 9.5 (x - 7)) (2 x + 1)$

Apply the distributive property.

$(- 1.9 x \cdot x - 1.9 x \cdot - 7 - 9.5 (x - 7)) (2 x + 1)$

Apply the distributive property.

$(- 1.9 x \cdot x - 1.9 x \cdot - 7 - 9.5 x - 9.5 \cdot - 7) (2 x + 1)$

Simplify and combine like terms.

Simplify each term.

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

Move $x$ .

$(- 1.9 (x \cdot x) - 1.9 x \cdot - 7 - 9.5 x - 9.5 \cdot - 7) (2 x + 1)$

Multiply $x$ by $x$ .

$(- 1.9 x^{2} - 1.9 x \cdot - 7 - 9.5 x - 9.5 \cdot - 7) (2 x + 1)$

Multiply $- 7$ by $- 1.9$ .

$(- 1.9 x^{2} + 13.3 x - 9.5 x - 9.5 \cdot - 7) (2 x + 1)$

Multiply $- 9.5$ by $- 7$ .

$(- 1.9 x^{2} + 13.3 x - 9.5 x + 66.5) (2 x + 1)$

Subtract $9.5 x$ from $13.3 x$ .

$(- 1.9 x^{2} + 3.8 x + 66.5) (2 x + 1)$

Expand $(- 1.9 x^{2} + 3.8 x + 66.5) (2 x + 1)$ by multiplying each term in the first expression by each term in the second expression.

$- 1.9 x^{2} (2 x) - 1.9 x^{2} \cdot 1 + 3.8 x (2 x) + 3.8 x \cdot 1 + 66.5 (2 x) + 66.5 \cdot 1$

Simplify terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$- 1.9 \cdot 2 x^{2} x - 1.9 x^{2} \cdot 1 + 3.8 x (2 x) + 3.8 x \cdot 1 + 66.5 (2 x) + 66.5 \cdot 1$

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

Move $x$ .

$- 1.9 \cdot 2 (x \cdot x^{2}) - 1.9 x^{2} \cdot 1 + 3.8 x (2 x) + 3.8 x \cdot 1 + 66.5 (2 x) + 66.5 \cdot 1$

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

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

$- 1.9 \cdot 2 (x^{1} x^{2}) - 1.9 x^{2} \cdot 1 + 3.8 x (2 x) + 3.8 x \cdot 1 + 66.5 (2 x) + 66.5 \cdot 1$

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

$- 1.9 \cdot 2 x^{1 + 2} - 1.9 x^{2} \cdot 1 + 3.8 x (2 x) + 3.8 x \cdot 1 + 66.5 (2 x) + 66.5 \cdot 1$

Add $1$ and $2$ .

$- 1.9 \cdot 2 x^{3} - 1.9 x^{2} \cdot 1 + 3.8 x (2 x) + 3.8 x \cdot 1 + 66.5 (2 x) + 66.5 \cdot 1$

Multiply $- 1.9$ by $2$ .

$- 3.8 x^{3} - 1.9 x^{2} \cdot 1 + 3.8 x (2 x) + 3.8 x \cdot 1 + 66.5 (2 x) + 66.5 \cdot 1$

Multiply $- 1.9$ by $1$ .

$- 3.8 x^{3} - 1.9 x^{2} + 3.8 x (2 x) + 3.8 x \cdot 1 + 66.5 (2 x) + 66.5 \cdot 1$

Rewrite using the commutative property of multiplication.

$- 3.8 x^{3} - 1.9 x^{2} + 3.8 \cdot 2 x \cdot x + 3.8 x \cdot 1 + 66.5 (2 x) + 66.5 \cdot 1$

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

Move $x$ .

$- 3.8 x^{3} - 1.9 x^{2} + 3.8 \cdot 2 (x \cdot x) + 3.8 x \cdot 1 + 66.5 (2 x) + 66.5 \cdot 1$

Multiply $x$ by $x$ .

$- 3.8 x^{3} - 1.9 x^{2} + 3.8 \cdot 2 x^{2} + 3.8 x \cdot 1 + 66.5 (2 x) + 66.5 \cdot 1$

Multiply $3.8$ by $2$ .

$- 3.8 x^{3} - 1.9 x^{2} + 7.6 x^{2} + 3.8 x \cdot 1 + 66.5 (2 x) + 66.5 \cdot 1$

Multiply $3.8$ by $1$ .

$- 3.8 x^{3} - 1.9 x^{2} + 7.6 x^{2} + 3.8 x + 66.5 (2 x) + 66.5 \cdot 1$

Multiply $2$ by $66.5$ .

$- 3.8 x^{3} - 1.9 x^{2} + 7.6 x^{2} + 3.8 x + 133 x + 66.5 \cdot 1$

Multiply $66.5$ by $1$ .

$- 3.8 x^{3} - 1.9 x^{2} + 7.6 x^{2} + 3.8 x + 133 x + 66.5$

Simplify by adding terms.

Add $- 1.9 x^{2}$ and $7.6 x^{2}$ .

$- 3.8 x^{3} + 5.7 x^{2} + 3.8 x + 133 x + 66.5$

Add $3.8 x$ and $133 x$ .

$- 3.8 x^{3} + 5.7 x^{2} + 136.8 x + 66.5$

The largest exponent is the degree of the polynomial.

$3$

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 by multiplying through.

Apply the distributive property.

$(- 1.9 x - 1.9 \cdot 5) (x - 7) (2 x + 1)$

Multiply $- 1.9$ by $5$ .

$(- 1.9 x - 9.5) (x - 7) (2 x + 1)$

Expand $(- 1.9 x - 9.5) (x - 7)$ using the FOIL Method.

Apply the distributive property.

$(- 1.9 x (x - 7) - 9.5 (x - 7)) (2 x + 1)$

Apply the distributive property.

$(- 1.9 x \cdot x - 1.9 x \cdot - 7 - 9.5 (x - 7)) (2 x + 1)$

Apply the distributive property.

$(- 1.9 x \cdot x - 1.9 x \cdot - 7 - 9.5 x - 9.5 \cdot - 7) (2 x + 1)$

Simplify and combine like terms.

Simplify each term.

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

Move $x$ .

$(- 1.9 (x \cdot x) - 1.9 x \cdot - 7 - 9.5 x - 9.5 \cdot - 7) (2 x + 1)$

Multiply $x$ by $x$ .

$(- 1.9 x^{2} - 1.9 x \cdot - 7 - 9.5 x - 9.5 \cdot - 7) (2 x + 1)$

Multiply $- 7$ by $- 1.9$ .

$(- 1.9 x^{2} + 13.3 x - 9.5 x - 9.5 \cdot - 7) (2 x + 1)$

Multiply $- 9.5$ by $- 7$ .

$(- 1.9 x^{2} + 13.3 x - 9.5 x + 66.5) (2 x + 1)$

Subtract $9.5 x$ from $13.3 x$ .

$(- 1.9 x^{2} + 3.8 x + 66.5) (2 x + 1)$

Expand $(- 1.9 x^{2} + 3.8 x + 66.5) (2 x + 1)$ by multiplying each term in the first expression by each term in the second expression.

$- 1.9 x^{2} (2 x) - 1.9 x^{2} \cdot 1 + 3.8 x (2 x) + 3.8 x \cdot 1 + 66.5 (2 x) + 66.5 \cdot 1$

Simplify terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$- 1.9 \cdot 2 x^{2} x - 1.9 x^{2} \cdot 1 + 3.8 x (2 x) + 3.8 x \cdot 1 + 66.5 (2 x) + 66.5 \cdot 1$

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

Move $x$ .

$- 1.9 \cdot 2 (x \cdot x^{2}) - 1.9 x^{2} \cdot 1 + 3.8 x (2 x) + 3.8 x \cdot 1 + 66.5 (2 x) + 66.5 \cdot 1$

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

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

$- 1.9 \cdot 2 (x^{1} x^{2}) - 1.9 x^{2} \cdot 1 + 3.8 x (2 x) + 3.8 x \cdot 1 + 66.5 (2 x) + 66.5 \cdot 1$

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

$- 1.9 \cdot 2 x^{1 + 2} - 1.9 x^{2} \cdot 1 + 3.8 x (2 x) + 3.8 x \cdot 1 + 66.5 (2 x) + 66.5 \cdot 1$

Add $1$ and $2$ .

$- 1.9 \cdot 2 x^{3} - 1.9 x^{2} \cdot 1 + 3.8 x (2 x) + 3.8 x \cdot 1 + 66.5 (2 x) + 66.5 \cdot 1$

Multiply $- 1.9$ by $2$ .

$- 3.8 x^{3} - 1.9 x^{2} \cdot 1 + 3.8 x (2 x) + 3.8 x \cdot 1 + 66.5 (2 x) + 66.5 \cdot 1$

Multiply $- 1.9$ by $1$ .

$- 3.8 x^{3} - 1.9 x^{2} + 3.8 x (2 x) + 3.8 x \cdot 1 + 66.5 (2 x) + 66.5 \cdot 1$

Rewrite using the commutative property of multiplication.

$- 3.8 x^{3} - 1.9 x^{2} + 3.8 \cdot 2 x \cdot x + 3.8 x \cdot 1 + 66.5 (2 x) + 66.5 \cdot 1$

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

Move $x$ .

$- 3.8 x^{3} - 1.9 x^{2} + 3.8 \cdot 2 (x \cdot x) + 3.8 x \cdot 1 + 66.5 (2 x) + 66.5 \cdot 1$

Multiply $x$ by $x$ .

$- 3.8 x^{3} - 1.9 x^{2} + 3.8 \cdot 2 x^{2} + 3.8 x \cdot 1 + 66.5 (2 x) + 66.5 \cdot 1$

Multiply $3.8$ by $2$ .

$- 3.8 x^{3} - 1.9 x^{2} + 7.6 x^{2} + 3.8 x \cdot 1 + 66.5 (2 x) + 66.5 \cdot 1$

Multiply $3.8$ by $1$ .

$- 3.8 x^{3} - 1.9 x^{2} + 7.6 x^{2} + 3.8 x + 66.5 (2 x) + 66.5 \cdot 1$

Multiply $2$ by $66.5$ .

$- 3.8 x^{3} - 1.9 x^{2} + 7.6 x^{2} + 3.8 x + 133 x + 66.5 \cdot 1$

Multiply $66.5$ by $1$ .

$- 3.8 x^{3} - 1.9 x^{2} + 7.6 x^{2} + 3.8 x + 133 x + 66.5$

Simplify by adding terms.

Add $- 1.9 x^{2}$ and $7.6 x^{2}$ .

$- 3.8 x^{3} + 5.7 x^{2} + 3.8 x + 133 x + 66.5$

Add $3.8 x$ and $133 x$ .

$- 3.8 x^{3} + 5.7 x^{2} + 136.8 x + 66.5$

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

$- 3.8 x^{3}$

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

$- 3.8$

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)=-1.9(x+5)(x-7)(2x+1)

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