Find the End Behavior f(x)=(2x+7)(3x-1)(x+5)(x-7)

$f (x) = (2 x + 7) (3 x - 1) (x + 5) (x - 7)$

Step 1

Identify the degree of the function.

Simplify and reorder the polynomial.

Expand $(2 x + 7) (3 x - 1)$ using the FOIL Method.

Apply the distributive property.

$(2 x (3 x - 1) + 7 (3 x - 1)) (x + 5) (x - 7)$

Apply the distributive property.

$(2 x (3 x) + 2 x \cdot - 1 + 7 (3 x - 1)) (x + 5) (x - 7)$

Apply the distributive property.

$(2 x (3 x) + 2 x \cdot - 1 + 7 (3 x) + 7 \cdot - 1) (x + 5) (x - 7)$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$(2 \cdot 3 x \cdot x + 2 x \cdot - 1 + 7 (3 x) + 7 \cdot - 1) (x + 5) (x - 7)$

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

Move $x$ .

$(2 \cdot 3 (x \cdot x) + 2 x \cdot - 1 + 7 (3 x) + 7 \cdot - 1) (x + 5) (x - 7)$

Multiply $x$ by $x$ .

$(2 \cdot 3 x^{2} + 2 x \cdot - 1 + 7 (3 x) + 7 \cdot - 1) (x + 5) (x - 7)$

Multiply $2$ by $3$ .

$(6 x^{2} + 2 x \cdot - 1 + 7 (3 x) + 7 \cdot - 1) (x + 5) (x - 7)$

Multiply $- 1$ by $2$ .

$(6 x^{2} - 2 x + 7 (3 x) + 7 \cdot - 1) (x + 5) (x - 7)$

Multiply $3$ by $7$ .

$(6 x^{2} - 2 x + 21 x + 7 \cdot - 1) (x + 5) (x - 7)$

Multiply $7$ by $- 1$ .

$(6 x^{2} - 2 x + 21 x - 7) (x + 5) (x - 7)$

Add $- 2 x$ and $21 x$ .

$(6 x^{2} + 19 x - 7) (x + 5) (x - 7)$

Expand $(6 x^{2} + 19 x - 7) (x + 5)$ by multiplying each term in the first expression by each term in the second expression.

$(6 x^{2} x + 6 x^{2} \cdot 5 + 19 x \cdot x + 19 x \cdot 5 - 7 x - 7 \cdot 5) (x - 7)$

Simplify terms.

Simplify each term.

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

Move $x$ .

$(6 (x \cdot x^{2}) + 6 x^{2} \cdot 5 + 19 x \cdot x + 19 x \cdot 5 - 7 x - 7 \cdot 5) (x - 7)$

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

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

$(6 (x^{1} x^{2}) + 6 x^{2} \cdot 5 + 19 x \cdot x + 19 x \cdot 5 - 7 x - 7 \cdot 5) (x - 7)$

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

$(6 x^{1 + 2} + 6 x^{2} \cdot 5 + 19 x \cdot x + 19 x \cdot 5 - 7 x - 7 \cdot 5) (x - 7)$

Add $1$ and $2$ .

$(6 x^{3} + 6 x^{2} \cdot 5 + 19 x \cdot x + 19 x \cdot 5 - 7 x - 7 \cdot 5) (x - 7)$

Multiply $5$ by $6$ .

$(6 x^{3} + 30 x^{2} + 19 x \cdot x + 19 x \cdot 5 - 7 x - 7 \cdot 5) (x - 7)$

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

Move $x$ .

$(6 x^{3} + 30 x^{2} + 19 (x \cdot x) + 19 x \cdot 5 - 7 x - 7 \cdot 5) (x - 7)$

Multiply $x$ by $x$ .

$(6 x^{3} + 30 x^{2} + 19 x^{2} + 19 x \cdot 5 - 7 x - 7 \cdot 5) (x - 7)$

Multiply $5$ by $19$ .

$(6 x^{3} + 30 x^{2} + 19 x^{2} + 95 x - 7 x - 7 \cdot 5) (x - 7)$

Multiply $- 7$ by $5$ .

$(6 x^{3} + 30 x^{2} + 19 x^{2} + 95 x - 7 x - 35) (x - 7)$

Simplify by adding terms.

Add $30 x^{2}$ and $19 x^{2}$ .

$(6 x^{3} + 49 x^{2} + 95 x - 7 x - 35) (x - 7)$

Subtract $7 x$ from $95 x$ .

$(6 x^{3} + 49 x^{2} + 88 x - 35) (x - 7)$

Expand $(6 x^{3} + 49 x^{2} + 88 x - 35) (x - 7)$ by multiplying each term in the first expression by each term in the second expression.

$6 x^{3} x + 6 x^{3} \cdot - 7 + 49 x^{2} x + 49 x^{2} \cdot - 7 + 88 x \cdot x + 88 x \cdot - 7 - 35 x - 35 \cdot - 7$

Simplify terms.

Simplify each term.

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

Move $x$ .

$6 (x \cdot x^{3}) + 6 x^{3} \cdot - 7 + 49 x^{2} x + 49 x^{2} \cdot - 7 + 88 x \cdot x + 88 x \cdot - 7 - 35 x - 35 \cdot - 7$

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

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

$6 (x^{1} x^{3}) + 6 x^{3} \cdot - 7 + 49 x^{2} x + 49 x^{2} \cdot - 7 + 88 x \cdot x + 88 x \cdot - 7 - 35 x - 35 \cdot - 7$

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

$6 x^{1 + 3} + 6 x^{3} \cdot - 7 + 49 x^{2} x + 49 x^{2} \cdot - 7 + 88 x \cdot x + 88 x \cdot - 7 - 35 x - 35 \cdot - 7$

Add $1$ and $3$ .

$6 x^{4} + 6 x^{3} \cdot - 7 + 49 x^{2} x + 49 x^{2} \cdot - 7 + 88 x \cdot x + 88 x \cdot - 7 - 35 x - 35 \cdot - 7$

Multiply $- 7$ by $6$ .

$6 x^{4} - 42 x^{3} + 49 x^{2} x + 49 x^{2} \cdot - 7 + 88 x \cdot x + 88 x \cdot - 7 - 35 x - 35 \cdot - 7$

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

Move $x$ .

$6 x^{4} - 42 x^{3} + 49 (x \cdot x^{2}) + 49 x^{2} \cdot - 7 + 88 x \cdot x + 88 x \cdot - 7 - 35 x - 35 \cdot - 7$

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

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

$6 x^{4} - 42 x^{3} + 49 (x^{1} x^{2}) + 49 x^{2} \cdot - 7 + 88 x \cdot x + 88 x \cdot - 7 - 35 x - 35 \cdot - 7$

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

$6 x^{4} - 42 x^{3} + 49 x^{1 + 2} + 49 x^{2} \cdot - 7 + 88 x \cdot x + 88 x \cdot - 7 - 35 x - 35 \cdot - 7$

Add $1$ and $2$ .

$6 x^{4} - 42 x^{3} + 49 x^{3} + 49 x^{2} \cdot - 7 + 88 x \cdot x + 88 x \cdot - 7 - 35 x - 35 \cdot - 7$

Multiply $- 7$ by $49$ .

$6 x^{4} - 42 x^{3} + 49 x^{3} - 343 x^{2} + 88 x \cdot x + 88 x \cdot - 7 - 35 x - 35 \cdot - 7$

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

Move $x$ .

$6 x^{4} - 42 x^{3} + 49 x^{3} - 343 x^{2} + 88 (x \cdot x) + 88 x \cdot - 7 - 35 x - 35 \cdot - 7$

Multiply $x$ by $x$ .

$6 x^{4} - 42 x^{3} + 49 x^{3} - 343 x^{2} + 88 x^{2} + 88 x \cdot - 7 - 35 x - 35 \cdot - 7$

Multiply $- 7$ by $88$ .

$6 x^{4} - 42 x^{3} + 49 x^{3} - 343 x^{2} + 88 x^{2} - 616 x - 35 x - 35 \cdot - 7$

Multiply $- 35$ by $- 7$ .

$6 x^{4} - 42 x^{3} + 49 x^{3} - 343 x^{2} + 88 x^{2} - 616 x - 35 x + 245$

Simplify by adding terms.

Add $- 42 x^{3}$ and $49 x^{3}$ .

$6 x^{4} + 7 x^{3} - 343 x^{2} + 88 x^{2} - 616 x - 35 x + 245$

Add $- 343 x^{2}$ and $88 x^{2}$ .

$6 x^{4} + 7 x^{3} - 255 x^{2} - 616 x - 35 x + 245$

Subtract $35 x$ from $- 616 x$ .

$6 x^{4} + 7 x^{3} - 255 x^{2} - 651 x + 245$

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.

Expand $(2 x + 7) (3 x - 1)$ using the FOIL Method.

Apply the distributive property.

$(2 x (3 x - 1) + 7 (3 x - 1)) (x + 5) (x - 7)$

Apply the distributive property.

$(2 x (3 x) + 2 x \cdot - 1 + 7 (3 x - 1)) (x + 5) (x - 7)$

Apply the distributive property.

$(2 x (3 x) + 2 x \cdot - 1 + 7 (3 x) + 7 \cdot - 1) (x + 5) (x - 7)$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$(2 \cdot 3 x \cdot x + 2 x \cdot - 1 + 7 (3 x) + 7 \cdot - 1) (x + 5) (x - 7)$

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

Move $x$ .

$(2 \cdot 3 (x \cdot x) + 2 x \cdot - 1 + 7 (3 x) + 7 \cdot - 1) (x + 5) (x - 7)$

Multiply $x$ by $x$ .

$(2 \cdot 3 x^{2} + 2 x \cdot - 1 + 7 (3 x) + 7 \cdot - 1) (x + 5) (x - 7)$

Multiply $2$ by $3$ .

$(6 x^{2} + 2 x \cdot - 1 + 7 (3 x) + 7 \cdot - 1) (x + 5) (x - 7)$

Multiply $- 1$ by $2$ .

$(6 x^{2} - 2 x + 7 (3 x) + 7 \cdot - 1) (x + 5) (x - 7)$

Multiply $3$ by $7$ .

$(6 x^{2} - 2 x + 21 x + 7 \cdot - 1) (x + 5) (x - 7)$

Multiply $7$ by $- 1$ .

$(6 x^{2} - 2 x + 21 x - 7) (x + 5) (x - 7)$

Add $- 2 x$ and $21 x$ .

$(6 x^{2} + 19 x - 7) (x + 5) (x - 7)$

Expand $(6 x^{2} + 19 x - 7) (x + 5)$ by multiplying each term in the first expression by each term in the second expression.

$(6 x^{2} x + 6 x^{2} \cdot 5 + 19 x \cdot x + 19 x \cdot 5 - 7 x - 7 \cdot 5) (x - 7)$

Simplify terms.

Simplify each term.

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

Move $x$ .

$(6 (x \cdot x^{2}) + 6 x^{2} \cdot 5 + 19 x \cdot x + 19 x \cdot 5 - 7 x - 7 \cdot 5) (x - 7)$

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

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

$(6 (x^{1} x^{2}) + 6 x^{2} \cdot 5 + 19 x \cdot x + 19 x \cdot 5 - 7 x - 7 \cdot 5) (x - 7)$

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

$(6 x^{1 + 2} + 6 x^{2} \cdot 5 + 19 x \cdot x + 19 x \cdot 5 - 7 x - 7 \cdot 5) (x - 7)$

Add $1$ and $2$ .

$(6 x^{3} + 6 x^{2} \cdot 5 + 19 x \cdot x + 19 x \cdot 5 - 7 x - 7 \cdot 5) (x - 7)$

Multiply $5$ by $6$ .

$(6 x^{3} + 30 x^{2} + 19 x \cdot x + 19 x \cdot 5 - 7 x - 7 \cdot 5) (x - 7)$

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

Move $x$ .

$(6 x^{3} + 30 x^{2} + 19 (x \cdot x) + 19 x \cdot 5 - 7 x - 7 \cdot 5) (x - 7)$

Multiply $x$ by $x$ .

$(6 x^{3} + 30 x^{2} + 19 x^{2} + 19 x \cdot 5 - 7 x - 7 \cdot 5) (x - 7)$

Multiply $5$ by $19$ .

$(6 x^{3} + 30 x^{2} + 19 x^{2} + 95 x - 7 x - 7 \cdot 5) (x - 7)$

Multiply $- 7$ by $5$ .

$(6 x^{3} + 30 x^{2} + 19 x^{2} + 95 x - 7 x - 35) (x - 7)$

Simplify by adding terms.

Add $30 x^{2}$ and $19 x^{2}$ .

$(6 x^{3} + 49 x^{2} + 95 x - 7 x - 35) (x - 7)$

Subtract $7 x$ from $95 x$ .

$(6 x^{3} + 49 x^{2} + 88 x - 35) (x - 7)$

Expand $(6 x^{3} + 49 x^{2} + 88 x - 35) (x - 7)$ by multiplying each term in the first expression by each term in the second expression.

$6 x^{3} x + 6 x^{3} \cdot - 7 + 49 x^{2} x + 49 x^{2} \cdot - 7 + 88 x \cdot x + 88 x \cdot - 7 - 35 x - 35 \cdot - 7$

Simplify terms.

Simplify each term.

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

Move $x$ .

$6 (x \cdot x^{3}) + 6 x^{3} \cdot - 7 + 49 x^{2} x + 49 x^{2} \cdot - 7 + 88 x \cdot x + 88 x \cdot - 7 - 35 x - 35 \cdot - 7$

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

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

$6 (x^{1} x^{3}) + 6 x^{3} \cdot - 7 + 49 x^{2} x + 49 x^{2} \cdot - 7 + 88 x \cdot x + 88 x \cdot - 7 - 35 x - 35 \cdot - 7$

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

$6 x^{1 + 3} + 6 x^{3} \cdot - 7 + 49 x^{2} x + 49 x^{2} \cdot - 7 + 88 x \cdot x + 88 x \cdot - 7 - 35 x - 35 \cdot - 7$

Add $1$ and $3$ .

$6 x^{4} + 6 x^{3} \cdot - 7 + 49 x^{2} x + 49 x^{2} \cdot - 7 + 88 x \cdot x + 88 x \cdot - 7 - 35 x - 35 \cdot - 7$

Multiply $- 7$ by $6$ .

$6 x^{4} - 42 x^{3} + 49 x^{2} x + 49 x^{2} \cdot - 7 + 88 x \cdot x + 88 x \cdot - 7 - 35 x - 35 \cdot - 7$

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

Move $x$ .

$6 x^{4} - 42 x^{3} + 49 (x \cdot x^{2}) + 49 x^{2} \cdot - 7 + 88 x \cdot x + 88 x \cdot - 7 - 35 x - 35 \cdot - 7$

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

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

$6 x^{4} - 42 x^{3} + 49 (x^{1} x^{2}) + 49 x^{2} \cdot - 7 + 88 x \cdot x + 88 x \cdot - 7 - 35 x - 35 \cdot - 7$

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

$6 x^{4} - 42 x^{3} + 49 x^{1 + 2} + 49 x^{2} \cdot - 7 + 88 x \cdot x + 88 x \cdot - 7 - 35 x - 35 \cdot - 7$

Add $1$ and $2$ .

$6 x^{4} - 42 x^{3} + 49 x^{3} + 49 x^{2} \cdot - 7 + 88 x \cdot x + 88 x \cdot - 7 - 35 x - 35 \cdot - 7$

Multiply $- 7$ by $49$ .

$6 x^{4} - 42 x^{3} + 49 x^{3} - 343 x^{2} + 88 x \cdot x + 88 x \cdot - 7 - 35 x - 35 \cdot - 7$

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

Move $x$ .

$6 x^{4} - 42 x^{3} + 49 x^{3} - 343 x^{2} + 88 (x \cdot x) + 88 x \cdot - 7 - 35 x - 35 \cdot - 7$

Multiply $x$ by $x$ .

$6 x^{4} - 42 x^{3} + 49 x^{3} - 343 x^{2} + 88 x^{2} + 88 x \cdot - 7 - 35 x - 35 \cdot - 7$

Multiply $- 7$ by $88$ .

$6 x^{4} - 42 x^{3} + 49 x^{3} - 343 x^{2} + 88 x^{2} - 616 x - 35 x - 35 \cdot - 7$

Multiply $- 35$ by $- 7$ .

$6 x^{4} - 42 x^{3} + 49 x^{3} - 343 x^{2} + 88 x^{2} - 616 x - 35 x + 245$

Simplify by adding terms.

Add $- 42 x^{3}$ and $49 x^{3}$ .

$6 x^{4} + 7 x^{3} - 343 x^{2} + 88 x^{2} - 616 x - 35 x + 245$

Add $- 343 x^{2}$ and $88 x^{2}$ .

$6 x^{4} + 7 x^{3} - 255 x^{2} - 616 x - 35 x + 245$

Subtract $35 x$ from $- 616 x$ .

$6 x^{4} + 7 x^{3} - 255 x^{2} - 651 x + 245$

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

$6 x^{4}$

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

$6$

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)=(2x+7)(3x-1)(x+5)(x-7)

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