Find the End Behavior f(x)=0.4(2x-9)(3x+1)(x-7)(x+9)

$f (x) = 0.4 (2 x - 9) (3 x + 1) (x - 7) (x + 9)$

Step 1

Identify the degree of the function.

Simplify and reorder the polynomial.

Simplify by multiplying through.

Apply the distributive property.

$(0.4 (2 x) + 0.4 \cdot - 9) (3 x + 1) (x - 7) (x + 9)$

Multiply.

Multiply $2$ by $0.4$ .

$(0.8 x + 0.4 \cdot - 9) (3 x + 1) (x - 7) (x + 9)$

Multiply $0.4$ by $- 9$ .

$(0.8 x - 3.6) (3 x + 1) (x - 7) (x + 9)$

Expand $(0.8 x - 3.6) (3 x + 1)$ using the FOIL Method.

Apply the distributive property.

$(0.8 x (3 x + 1) - 3.6 (3 x + 1)) (x - 7) (x + 9)$

Apply the distributive property.

$(0.8 x (3 x) + 0.8 x \cdot 1 - 3.6 (3 x + 1)) (x - 7) (x + 9)$

Apply the distributive property.

$(0.8 x (3 x) + 0.8 x \cdot 1 - 3.6 (3 x) - 3.6 \cdot 1) (x - 7) (x + 9)$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$(0.8 \cdot 3 x \cdot x + 0.8 x \cdot 1 - 3.6 (3 x) - 3.6 \cdot 1) (x - 7) (x + 9)$

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

Move $x$ .

$(0.8 \cdot 3 (x \cdot x) + 0.8 x \cdot 1 - 3.6 (3 x) - 3.6 \cdot 1) (x - 7) (x + 9)$

Multiply $x$ by $x$ .

$(0.8 \cdot 3 x^{2} + 0.8 x \cdot 1 - 3.6 (3 x) - 3.6 \cdot 1) (x - 7) (x + 9)$

Multiply $0.8$ by $3$ .

$(2.4 x^{2} + 0.8 x \cdot 1 - 3.6 (3 x) - 3.6 \cdot 1) (x - 7) (x + 9)$

Multiply $0.8$ by $1$ .

$(2.4 x^{2} + 0.8 x - 3.6 (3 x) - 3.6 \cdot 1) (x - 7) (x + 9)$

Multiply $3$ by $- 3.6$ .

$(2.4 x^{2} + 0.8 x - 10.8 x - 3.6 \cdot 1) (x - 7) (x + 9)$

Multiply $- 3.6$ by $1$ .

$(2.4 x^{2} + 0.8 x - 10.8 x - 3.6) (x - 7) (x + 9)$

Subtract $10.8 x$ from $0.8 x$ .

$(2.4 x^{2} - 10 x - 3.6) (x - 7) (x + 9)$

Expand $(2.4 x^{2} - 10 x - 3.6) (x - 7)$ by multiplying each term in the first expression by each term in the second expression.

$(2.4 x^{2} x + 2.4 x^{2} \cdot - 7 - 10 x \cdot x - 10 x \cdot - 7 - 3.6 x - 3.6 \cdot - 7) (x + 9)$

Simplify terms.

Simplify each term.

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

Move $x$ .

$(2.4 (x \cdot x^{2}) + 2.4 x^{2} \cdot - 7 - 10 x \cdot x - 10 x \cdot - 7 - 3.6 x - 3.6 \cdot - 7) (x + 9)$

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

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

$(2.4 (x^{1} x^{2}) + 2.4 x^{2} \cdot - 7 - 10 x \cdot x - 10 x \cdot - 7 - 3.6 x - 3.6 \cdot - 7) (x + 9)$

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

$(2.4 x^{1 + 2} + 2.4 x^{2} \cdot - 7 - 10 x \cdot x - 10 x \cdot - 7 - 3.6 x - 3.6 \cdot - 7) (x + 9)$

Add $1$ and $2$ .

$(2.4 x^{3} + 2.4 x^{2} \cdot - 7 - 10 x \cdot x - 10 x \cdot - 7 - 3.6 x - 3.6 \cdot - 7) (x + 9)$

Multiply $- 7$ by $2.4$ .

$(2.4 x^{3} - 16.8 x^{2} - 10 x \cdot x - 10 x \cdot - 7 - 3.6 x - 3.6 \cdot - 7) (x + 9)$

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

Move $x$ .

$(2.4 x^{3} - 16.8 x^{2} - 10 (x \cdot x) - 10 x \cdot - 7 - 3.6 x - 3.6 \cdot - 7) (x + 9)$

Multiply $x$ by $x$ .

$(2.4 x^{3} - 16.8 x^{2} - 10 x^{2} - 10 x \cdot - 7 - 3.6 x - 3.6 \cdot - 7) (x + 9)$

Multiply $- 7$ by $- 10$ .

$(2.4 x^{3} - 16.8 x^{2} - 10 x^{2} + 70 x - 3.6 x - 3.6 \cdot - 7) (x + 9)$

Multiply $- 3.6$ by $- 7$ .

$(2.4 x^{3} - 16.8 x^{2} - 10 x^{2} + 70 x - 3.6 x + 25.2) (x + 9)$

Simplify by adding terms.

Subtract $10 x^{2}$ from $- 16.8 x^{2}$ .

$(2.4 x^{3} - 26.8 x^{2} + 70 x - 3.6 x + 25.2) (x + 9)$

Subtract $3.6 x$ from $70 x$ .

$(2.4 x^{3} - 26.8 x^{2} + 66.4 x + 25.2) (x + 9)$

Expand $(2.4 x^{3} - 26.8 x^{2} + 66.4 x + 25.2) (x + 9)$ by multiplying each term in the first expression by each term in the second expression.

$2.4 x^{3} x + 2.4 x^{3} \cdot 9 - 26.8 x^{2} x - 26.8 x^{2} \cdot 9 + 66.4 x \cdot x + 66.4 x \cdot 9 + 25.2 x + 25.2 \cdot 9$

Simplify terms.

Simplify each term.

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

Move $x$ .

$2.4 (x \cdot x^{3}) + 2.4 x^{3} \cdot 9 - 26.8 x^{2} x - 26.8 x^{2} \cdot 9 + 66.4 x \cdot x + 66.4 x \cdot 9 + 25.2 x + 25.2 \cdot 9$

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

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

$2.4 (x^{1} x^{3}) + 2.4 x^{3} \cdot 9 - 26.8 x^{2} x - 26.8 x^{2} \cdot 9 + 66.4 x \cdot x + 66.4 x \cdot 9 + 25.2 x + 25.2 \cdot 9$

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

$2.4 x^{1 + 3} + 2.4 x^{3} \cdot 9 - 26.8 x^{2} x - 26.8 x^{2} \cdot 9 + 66.4 x \cdot x + 66.4 x \cdot 9 + 25.2 x + 25.2 \cdot 9$

Add $1$ and $3$ .

$2.4 x^{4} + 2.4 x^{3} \cdot 9 - 26.8 x^{2} x - 26.8 x^{2} \cdot 9 + 66.4 x \cdot x + 66.4 x \cdot 9 + 25.2 x + 25.2 \cdot 9$

Multiply $9$ by $2.4$ .

$2.4 x^{4} + 21.6 x^{3} - 26.8 x^{2} x - 26.8 x^{2} \cdot 9 + 66.4 x \cdot x + 66.4 x \cdot 9 + 25.2 x + 25.2 \cdot 9$

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

Move $x$ .

$2.4 x^{4} + 21.6 x^{3} - 26.8 (x \cdot x^{2}) - 26.8 x^{2} \cdot 9 + 66.4 x \cdot x + 66.4 x \cdot 9 + 25.2 x + 25.2 \cdot 9$

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

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

$2.4 x^{4} + 21.6 x^{3} - 26.8 (x^{1} x^{2}) - 26.8 x^{2} \cdot 9 + 66.4 x \cdot x + 66.4 x \cdot 9 + 25.2 x + 25.2 \cdot 9$

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

$2.4 x^{4} + 21.6 x^{3} - 26.8 x^{1 + 2} - 26.8 x^{2} \cdot 9 + 66.4 x \cdot x + 66.4 x \cdot 9 + 25.2 x + 25.2 \cdot 9$

Add $1$ and $2$ .

$2.4 x^{4} + 21.6 x^{3} - 26.8 x^{3} - 26.8 x^{2} \cdot 9 + 66.4 x \cdot x + 66.4 x \cdot 9 + 25.2 x + 25.2 \cdot 9$

Multiply $9$ by $- 26.8$ .

$2.4 x^{4} + 21.6 x^{3} - 26.8 x^{3} - 241.2 x^{2} + 66.4 x \cdot x + 66.4 x \cdot 9 + 25.2 x + 25.2 \cdot 9$

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

Move $x$ .

$2.4 x^{4} + 21.6 x^{3} - 26.8 x^{3} - 241.2 x^{2} + 66.4 (x \cdot x) + 66.4 x \cdot 9 + 25.2 x + 25.2 \cdot 9$

Multiply $x$ by $x$ .

$2.4 x^{4} + 21.6 x^{3} - 26.8 x^{3} - 241.2 x^{2} + 66.4 x^{2} + 66.4 x \cdot 9 + 25.2 x + 25.2 \cdot 9$

Multiply $9$ by $66.4$ .

$2.4 x^{4} + 21.6 x^{3} - 26.8 x^{3} - 241.2 x^{2} + 66.4 x^{2} + 597.6 x + 25.2 x + 25.2 \cdot 9$

Multiply $25.2$ by $9$ .

$2.4 x^{4} + 21.6 x^{3} - 26.8 x^{3} - 241.2 x^{2} + 66.4 x^{2} + 597.6 x + 25.2 x + 226.8$

Simplify by adding terms.

Subtract $26.8 x^{3}$ from $21.6 x^{3}$ .

$2.4 x^{4} - 5.2 x^{3} - 241.2 x^{2} + 66.4 x^{2} + 597.6 x + 25.2 x + 226.8$

Add $- 241.2 x^{2}$ and $66.4 x^{2}$ .

$2.4 x^{4} - 5.2 x^{3} - 174.8 x^{2} + 597.6 x + 25.2 x + 226.8$

Add $597.6 x$ and $25.2 x$ .

$2.4 x^{4} - 5.2 x^{3} - 174.8 x^{2} + 622.8 x + 226.8$

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 by multiplying through.

Apply the distributive property.

$(0.4 (2 x) + 0.4 \cdot - 9) (3 x + 1) (x - 7) (x + 9)$

Multiply.

Multiply $2$ by $0.4$ .

$(0.8 x + 0.4 \cdot - 9) (3 x + 1) (x - 7) (x + 9)$

Multiply $0.4$ by $- 9$ .

$(0.8 x - 3.6) (3 x + 1) (x - 7) (x + 9)$

Expand $(0.8 x - 3.6) (3 x + 1)$ using the FOIL Method.

Apply the distributive property.

$(0.8 x (3 x + 1) - 3.6 (3 x + 1)) (x - 7) (x + 9)$

Apply the distributive property.

$(0.8 x (3 x) + 0.8 x \cdot 1 - 3.6 (3 x + 1)) (x - 7) (x + 9)$

Apply the distributive property.

$(0.8 x (3 x) + 0.8 x \cdot 1 - 3.6 (3 x) - 3.6 \cdot 1) (x - 7) (x + 9)$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$(0.8 \cdot 3 x \cdot x + 0.8 x \cdot 1 - 3.6 (3 x) - 3.6 \cdot 1) (x - 7) (x + 9)$

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

Move $x$ .

$(0.8 \cdot 3 (x \cdot x) + 0.8 x \cdot 1 - 3.6 (3 x) - 3.6 \cdot 1) (x - 7) (x + 9)$

Multiply $x$ by $x$ .

$(0.8 \cdot 3 x^{2} + 0.8 x \cdot 1 - 3.6 (3 x) - 3.6 \cdot 1) (x - 7) (x + 9)$

Multiply $0.8$ by $3$ .

$(2.4 x^{2} + 0.8 x \cdot 1 - 3.6 (3 x) - 3.6 \cdot 1) (x - 7) (x + 9)$

Multiply $0.8$ by $1$ .

$(2.4 x^{2} + 0.8 x - 3.6 (3 x) - 3.6 \cdot 1) (x - 7) (x + 9)$

Multiply $3$ by $- 3.6$ .

$(2.4 x^{2} + 0.8 x - 10.8 x - 3.6 \cdot 1) (x - 7) (x + 9)$

Multiply $- 3.6$ by $1$ .

$(2.4 x^{2} + 0.8 x - 10.8 x - 3.6) (x - 7) (x + 9)$

Subtract $10.8 x$ from $0.8 x$ .

$(2.4 x^{2} - 10 x - 3.6) (x - 7) (x + 9)$

Expand $(2.4 x^{2} - 10 x - 3.6) (x - 7)$ by multiplying each term in the first expression by each term in the second expression.

$(2.4 x^{2} x + 2.4 x^{2} \cdot - 7 - 10 x \cdot x - 10 x \cdot - 7 - 3.6 x - 3.6 \cdot - 7) (x + 9)$

Simplify terms.

Simplify each term.

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

Move $x$ .

$(2.4 (x \cdot x^{2}) + 2.4 x^{2} \cdot - 7 - 10 x \cdot x - 10 x \cdot - 7 - 3.6 x - 3.6 \cdot - 7) (x + 9)$

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

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

$(2.4 (x^{1} x^{2}) + 2.4 x^{2} \cdot - 7 - 10 x \cdot x - 10 x \cdot - 7 - 3.6 x - 3.6 \cdot - 7) (x + 9)$

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

$(2.4 x^{1 + 2} + 2.4 x^{2} \cdot - 7 - 10 x \cdot x - 10 x \cdot - 7 - 3.6 x - 3.6 \cdot - 7) (x + 9)$

Add $1$ and $2$ .

$(2.4 x^{3} + 2.4 x^{2} \cdot - 7 - 10 x \cdot x - 10 x \cdot - 7 - 3.6 x - 3.6 \cdot - 7) (x + 9)$

Multiply $- 7$ by $2.4$ .

$(2.4 x^{3} - 16.8 x^{2} - 10 x \cdot x - 10 x \cdot - 7 - 3.6 x - 3.6 \cdot - 7) (x + 9)$

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

Move $x$ .

$(2.4 x^{3} - 16.8 x^{2} - 10 (x \cdot x) - 10 x \cdot - 7 - 3.6 x - 3.6 \cdot - 7) (x + 9)$

Multiply $x$ by $x$ .

$(2.4 x^{3} - 16.8 x^{2} - 10 x^{2} - 10 x \cdot - 7 - 3.6 x - 3.6 \cdot - 7) (x + 9)$

Multiply $- 7$ by $- 10$ .

$(2.4 x^{3} - 16.8 x^{2} - 10 x^{2} + 70 x - 3.6 x - 3.6 \cdot - 7) (x + 9)$

Multiply $- 3.6$ by $- 7$ .

$(2.4 x^{3} - 16.8 x^{2} - 10 x^{2} + 70 x - 3.6 x + 25.2) (x + 9)$

Simplify by adding terms.

Subtract $10 x^{2}$ from $- 16.8 x^{2}$ .

$(2.4 x^{3} - 26.8 x^{2} + 70 x - 3.6 x + 25.2) (x + 9)$

Subtract $3.6 x$ from $70 x$ .

$(2.4 x^{3} - 26.8 x^{2} + 66.4 x + 25.2) (x + 9)$

Expand $(2.4 x^{3} - 26.8 x^{2} + 66.4 x + 25.2) (x + 9)$ by multiplying each term in the first expression by each term in the second expression.

$2.4 x^{3} x + 2.4 x^{3} \cdot 9 - 26.8 x^{2} x - 26.8 x^{2} \cdot 9 + 66.4 x \cdot x + 66.4 x \cdot 9 + 25.2 x + 25.2 \cdot 9$

Simplify terms.

Simplify each term.

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

Move $x$ .

$2.4 (x \cdot x^{3}) + 2.4 x^{3} \cdot 9 - 26.8 x^{2} x - 26.8 x^{2} \cdot 9 + 66.4 x \cdot x + 66.4 x \cdot 9 + 25.2 x + 25.2 \cdot 9$

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

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

$2.4 (x^{1} x^{3}) + 2.4 x^{3} \cdot 9 - 26.8 x^{2} x - 26.8 x^{2} \cdot 9 + 66.4 x \cdot x + 66.4 x \cdot 9 + 25.2 x + 25.2 \cdot 9$

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

$2.4 x^{1 + 3} + 2.4 x^{3} \cdot 9 - 26.8 x^{2} x - 26.8 x^{2} \cdot 9 + 66.4 x \cdot x + 66.4 x \cdot 9 + 25.2 x + 25.2 \cdot 9$

Add $1$ and $3$ .

$2.4 x^{4} + 2.4 x^{3} \cdot 9 - 26.8 x^{2} x - 26.8 x^{2} \cdot 9 + 66.4 x \cdot x + 66.4 x \cdot 9 + 25.2 x + 25.2 \cdot 9$

Multiply $9$ by $2.4$ .

$2.4 x^{4} + 21.6 x^{3} - 26.8 x^{2} x - 26.8 x^{2} \cdot 9 + 66.4 x \cdot x + 66.4 x \cdot 9 + 25.2 x + 25.2 \cdot 9$

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

Move $x$ .

$2.4 x^{4} + 21.6 x^{3} - 26.8 (x \cdot x^{2}) - 26.8 x^{2} \cdot 9 + 66.4 x \cdot x + 66.4 x \cdot 9 + 25.2 x + 25.2 \cdot 9$

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

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

$2.4 x^{4} + 21.6 x^{3} - 26.8 (x^{1} x^{2}) - 26.8 x^{2} \cdot 9 + 66.4 x \cdot x + 66.4 x \cdot 9 + 25.2 x + 25.2 \cdot 9$

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

$2.4 x^{4} + 21.6 x^{3} - 26.8 x^{1 + 2} - 26.8 x^{2} \cdot 9 + 66.4 x \cdot x + 66.4 x \cdot 9 + 25.2 x + 25.2 \cdot 9$

Add $1$ and $2$ .

$2.4 x^{4} + 21.6 x^{3} - 26.8 x^{3} - 26.8 x^{2} \cdot 9 + 66.4 x \cdot x + 66.4 x \cdot 9 + 25.2 x + 25.2 \cdot 9$

Multiply $9$ by $- 26.8$ .

$2.4 x^{4} + 21.6 x^{3} - 26.8 x^{3} - 241.2 x^{2} + 66.4 x \cdot x + 66.4 x \cdot 9 + 25.2 x + 25.2 \cdot 9$

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

Move $x$ .

$2.4 x^{4} + 21.6 x^{3} - 26.8 x^{3} - 241.2 x^{2} + 66.4 (x \cdot x) + 66.4 x \cdot 9 + 25.2 x + 25.2 \cdot 9$

Multiply $x$ by $x$ .

$2.4 x^{4} + 21.6 x^{3} - 26.8 x^{3} - 241.2 x^{2} + 66.4 x^{2} + 66.4 x \cdot 9 + 25.2 x + 25.2 \cdot 9$

Multiply $9$ by $66.4$ .

$2.4 x^{4} + 21.6 x^{3} - 26.8 x^{3} - 241.2 x^{2} + 66.4 x^{2} + 597.6 x + 25.2 x + 25.2 \cdot 9$

Multiply $25.2$ by $9$ .

$2.4 x^{4} + 21.6 x^{3} - 26.8 x^{3} - 241.2 x^{2} + 66.4 x^{2} + 597.6 x + 25.2 x + 226.8$

Simplify by adding terms.

Subtract $26.8 x^{3}$ from $21.6 x^{3}$ .

$2.4 x^{4} - 5.2 x^{3} - 241.2 x^{2} + 66.4 x^{2} + 597.6 x + 25.2 x + 226.8$

Add $- 241.2 x^{2}$ and $66.4 x^{2}$ .

$2.4 x^{4} - 5.2 x^{3} - 174.8 x^{2} + 597.6 x + 25.2 x + 226.8$

Add $597.6 x$ and $25.2 x$ .

$2.4 x^{4} - 5.2 x^{3} - 174.8 x^{2} + 622.8 x + 226.8$

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

$2.4 x^{4}$

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

$2.4$

Step 4

Since the leading coefficient is positive, the graph rises to the right.

Positive

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 rises to the right

Step 7

Find the End Behavior f(x)=0.4(2x-9)(3x+1)(x-7)(x+9)

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