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

$f (x) = (x - 2) {(x + 1)}^{2} (x - 5)$

Step 1

Identify the degree of the function.

Simplify and reorder the polynomial.

Rewrite ${(x + 1)}^{2}$ as $(x + 1) (x + 1)$ .

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

Expand $(x + 1) (x + 1)$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

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

Multiply $x$ by $1$ .

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

Multiply $x$ by $1$ .

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

Multiply $1$ by $1$ .

$(x - 2) (x^{2} + x + x + 1) (x - 5)$

Add $x$ and $x$ .

$(x - 2) (x^{2} + 2 x + 1) (x - 5)$

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

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

Simplify terms.

Simplify each term.

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

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

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

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

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

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

Add $1$ and $2$ .

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

Rewrite using the commutative property of multiplication.

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

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

Move $x$ .

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

Multiply $x$ by $x$ .

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

Multiply $x$ by $1$ .

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

Multiply $2$ by $- 2$ .

$(x^{3} + 2 x^{2} + x - 2 x^{2} - 4 x - 2 \cdot 1) (x - 5)$

Multiply $- 2$ by $1$ .

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

Simplify by adding terms.

Combine the opposite terms in $x^{3} + 2 x^{2} + x - 2 x^{2} - 4 x - 2$ .

Subtract $2 x^{2}$ from $2 x^{2}$ .

$(x^{3} + x + 0 - 4 x - 2) (x - 5)$

Add $x^{3} + x$ and $0$ .

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

Subtract $4 x$ from $x$ .

$(x^{3} - 3 x - 2) (x - 5)$

Expand $(x^{3} - 3 x - 2) (x - 5)$ by multiplying each term in the first expression by each term in the second expression.

$x^{3} x + x^{3} \cdot - 5 - 3 x \cdot x - 3 x \cdot - 5 - 2 x - 2 \cdot - 5$

Simplify terms.

Simplify each term.

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

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

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

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

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

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

Add $3$ and $1$ .

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

Move $- 5$ to the left of $x^{3}$ .

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

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

Move $x$ .

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

Multiply $x$ by $x$ .

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

Multiply $- 5$ by $- 3$ .

$x^{4} - 5 x^{3} - 3 x^{2} + 15 x - 2 x - 2 \cdot - 5$

Multiply $- 2$ by $- 5$ .

$x^{4} - 5 x^{3} - 3 x^{2} + 15 x - 2 x + 10$

Subtract $2 x$ from $15 x$ .

$x^{4} - 5 x^{3} - 3 x^{2} + 13 x + 10$

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.

Rewrite ${(x + 1)}^{2}$ as $(x + 1) (x + 1)$ .

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

Expand $(x + 1) (x + 1)$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

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

Multiply $x$ by $1$ .

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

Multiply $x$ by $1$ .

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

Multiply $1$ by $1$ .

$(x - 2) (x^{2} + x + x + 1) (x - 5)$

Add $x$ and $x$ .

$(x - 2) (x^{2} + 2 x + 1) (x - 5)$

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

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

Simplify terms.

Simplify each term.

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

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

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

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

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

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

Add $1$ and $2$ .

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

Rewrite using the commutative property of multiplication.

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

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

Move $x$ .

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

Multiply $x$ by $x$ .

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

Multiply $x$ by $1$ .

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

Multiply $2$ by $- 2$ .

$(x^{3} + 2 x^{2} + x - 2 x^{2} - 4 x - 2 \cdot 1) (x - 5)$

Multiply $- 2$ by $1$ .

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

Simplify by adding terms.

Combine the opposite terms in $x^{3} + 2 x^{2} + x - 2 x^{2} - 4 x - 2$ .

Subtract $2 x^{2}$ from $2 x^{2}$ .

$(x^{3} + x + 0 - 4 x - 2) (x - 5)$

Add $x^{3} + x$ and $0$ .

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

Subtract $4 x$ from $x$ .

$(x^{3} - 3 x - 2) (x - 5)$

Expand $(x^{3} - 3 x - 2) (x - 5)$ by multiplying each term in the first expression by each term in the second expression.

$x^{3} x + x^{3} \cdot - 5 - 3 x \cdot x - 3 x \cdot - 5 - 2 x - 2 \cdot - 5$

Simplify terms.

Simplify each term.

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

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

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

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

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

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

Add $3$ and $1$ .

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

Move $- 5$ to the left of $x^{3}$ .

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

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

Move $x$ .

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

Multiply $x$ by $x$ .

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

Multiply $- 5$ by $- 3$ .

$x^{4} - 5 x^{3} - 3 x^{2} + 15 x - 2 x - 2 \cdot - 5$

Multiply $- 2$ by $- 5$ .

$x^{4} - 5 x^{3} - 3 x^{2} + 15 x - 2 x + 10$

Subtract $2 x$ from $15 x$ .

$x^{4} - 5 x^{3} - 3 x^{2} + 13 x + 10$

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

$x^{4}$

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

$1$

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)=(x-2)(x+1)^2(x-5)

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