Find the End Behavior f(x)=(x-1)^3(x+1)

$f (x) = {(x - 1)}^{3} (x + 1)$

Step 1

Identify the degree of the function.

Simplify and reorder the polynomial.

Use the Binomial Theorem.

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

Simplify each term.

Multiply $- 1$ by $3$ .

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

Raise $- 1$ to the power of $2$ .

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

Multiply $3$ by $1$ .

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

Raise $- 1$ to the power of $3$ .

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

Expand $(x^{3} - 3 x^{2} + 3 x - 1) (x + 1)$ by multiplying each term in the first expression by each term in the second expression.

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

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 $1$ .

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

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

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

Add $3$ and $1$ .

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

Multiply $x^{3}$ by $1$ .

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

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

Move $x$ .

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

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

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

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

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

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

Add $1$ and $2$ .

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

Multiply $- 3$ by $1$ .

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

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

Move $x$ .

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

Multiply $x$ by $x$ .

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

Multiply $3$ by $1$ .

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

Rewrite $- 1 x$ as $- x$ .

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

Multiply $- 1$ by $1$ .

$x^{4} + x^{3} - 3 x^{3} - 3 x^{2} + 3 x^{2} + 3 x - x - 1$

Simplify by adding terms.

Combine the opposite terms in $x^{4} + x^{3} - 3 x^{3} - 3 x^{2} + 3 x^{2} + 3 x - x - 1$ .

Add $- 3 x^{2}$ and $3 x^{2}$ .

$x^{4} + x^{3} - 3 x^{3} + 0 + 3 x - x - 1$

Add $x^{4} + x^{3} - 3 x^{3}$ and $0$ .

$x^{4} + x^{3} - 3 x^{3} + 3 x - x - 1$

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

$x^{4} - 2 x^{3} + 3 x - x - 1$

Subtract $x$ from $3 x$ .

$x^{4} - 2 x^{3} + 2 x - 1$

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.

Use the Binomial Theorem.

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

Simplify each term.

Multiply $- 1$ by $3$ .

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

Raise $- 1$ to the power of $2$ .

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

Multiply $3$ by $1$ .

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

Raise $- 1$ to the power of $3$ .

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

Expand $(x^{3} - 3 x^{2} + 3 x - 1) (x + 1)$ by multiplying each term in the first expression by each term in the second expression.

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

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 $1$ .

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

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

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

Add $3$ and $1$ .

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

Multiply $x^{3}$ by $1$ .

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

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

Move $x$ .

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

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

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

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

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

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

Add $1$ and $2$ .

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

Multiply $- 3$ by $1$ .

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

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

Move $x$ .

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

Multiply $x$ by $x$ .

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

Multiply $3$ by $1$ .

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

Rewrite $- 1 x$ as $- x$ .

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

Multiply $- 1$ by $1$ .

$x^{4} + x^{3} - 3 x^{3} - 3 x^{2} + 3 x^{2} + 3 x - x - 1$

Simplify by adding terms.

Combine the opposite terms in $x^{4} + x^{3} - 3 x^{3} - 3 x^{2} + 3 x^{2} + 3 x - x - 1$ .

Add $- 3 x^{2}$ and $3 x^{2}$ .

$x^{4} + x^{3} - 3 x^{3} + 0 + 3 x - x - 1$

Add $x^{4} + x^{3} - 3 x^{3}$ and $0$ .

$x^{4} + x^{3} - 3 x^{3} + 3 x - x - 1$

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

$x^{4} - 2 x^{3} + 3 x - x - 1$

Subtract $x$ from $3 x$ .

$x^{4} - 2 x^{3} + 2 x - 1$

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-1)^3(x+1)

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