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

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

Step 1

Identify the degree of the function.

Simplify and reorder the polynomial.

Use the Binomial Theorem.

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

Simplify terms.

Simplify each term.

Multiply $- 1$ by $3$ .

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

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

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

Multiply $3$ by $1$ .

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

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

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

Apply the distributive property.

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

Simplify.

Multiply $x^{2}$ by $x^{3}$ by adding the exponents.

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

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

Add $2$ and $3$ .

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

Rewrite using the commutative property of multiplication.

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

Rewrite using the commutative property of multiplication.

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

Move $- 1$ to the left of $x^{2}$ .

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

Simplify each term.

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

Move $x^{2}$ .

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

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

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

Add $2$ and $2$ .

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

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

Move $x$ .

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

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

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

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

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

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

Add $1$ and $2$ .

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

Rewrite $- 1 x^{2}$ as $- x^{2}$ .

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

Expand $(x^{5} - 3 x^{4} + 3 x^{3} - x^{2}) (x + 1)$ by multiplying each term in the first expression by each term in the second expression.

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

Simplify terms.

Simplify each term.

Multiply $x^{5}$ by $x$ by adding the exponents.

Multiply $x^{5}$ by $x$ .

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

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

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

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

Add $5$ and $1$ .

$x^{6} + x^{5} \cdot 1 - 3 x^{4} x - 3 x^{4} \cdot 1 + 3 x^{3} x + 3 x^{3} \cdot 1 - x^{2} x - x^{2} \cdot 1$

Multiply $x^{5}$ by $1$ .

$x^{6} + x^{5} - 3 x^{4} x - 3 x^{4} \cdot 1 + 3 x^{3} x + 3 x^{3} \cdot 1 - x^{2} x - x^{2} \cdot 1$

Multiply $x^{4}$ by $x$ by adding the exponents.

Move $x$ .

$x^{6} + x^{5} - 3 (x \cdot x^{4}) - 3 x^{4} \cdot 1 + 3 x^{3} x + 3 x^{3} \cdot 1 - x^{2} x - x^{2} \cdot 1$

Multiply $x$ by $x^{4}$ .

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

$x^{6} + x^{5} - 3 (x^{1} x^{4}) - 3 x^{4} \cdot 1 + 3 x^{3} x + 3 x^{3} \cdot 1 - x^{2} x - x^{2} \cdot 1$

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

$x^{6} + x^{5} - 3 x^{1 + 4} - 3 x^{4} \cdot 1 + 3 x^{3} x + 3 x^{3} \cdot 1 - x^{2} x - x^{2} \cdot 1$

Add $1$ and $4$ .

$x^{6} + x^{5} - 3 x^{5} - 3 x^{4} \cdot 1 + 3 x^{3} x + 3 x^{3} \cdot 1 - x^{2} x - x^{2} \cdot 1$

Multiply $- 3$ by $1$ .

$x^{6} + x^{5} - 3 x^{5} - 3 x^{4} + 3 x^{3} x + 3 x^{3} \cdot 1 - x^{2} x - x^{2} \cdot 1$

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

Move $x$ .

$x^{6} + x^{5} - 3 x^{5} - 3 x^{4} + 3 (x \cdot x^{3}) + 3 x^{3} \cdot 1 - x^{2} x - x^{2} \cdot 1$

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

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

$x^{6} + x^{5} - 3 x^{5} - 3 x^{4} + 3 (x^{1} x^{3}) + 3 x^{3} \cdot 1 - x^{2} x - x^{2} \cdot 1$

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

$x^{6} + x^{5} - 3 x^{5} - 3 x^{4} + 3 x^{1 + 3} + 3 x^{3} \cdot 1 - x^{2} x - x^{2} \cdot 1$

Add $1$ and $3$ .

$x^{6} + x^{5} - 3 x^{5} - 3 x^{4} + 3 x^{4} + 3 x^{3} \cdot 1 - x^{2} x - x^{2} \cdot 1$

Multiply $3$ by $1$ .

$x^{6} + x^{5} - 3 x^{5} - 3 x^{4} + 3 x^{4} + 3 x^{3} - x^{2} x - x^{2} \cdot 1$

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

Move $x$ .

$x^{6} + x^{5} - 3 x^{5} - 3 x^{4} + 3 x^{4} + 3 x^{3} - (x \cdot x^{2}) - x^{2} \cdot 1$

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

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

$x^{6} + x^{5} - 3 x^{5} - 3 x^{4} + 3 x^{4} + 3 x^{3} - (x^{1} x^{2}) - x^{2} \cdot 1$

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

$x^{6} + x^{5} - 3 x^{5} - 3 x^{4} + 3 x^{4} + 3 x^{3} - x^{1 + 2} - x^{2} \cdot 1$

Add $1$ and $2$ .

$x^{6} + x^{5} - 3 x^{5} - 3 x^{4} + 3 x^{4} + 3 x^{3} - x^{3} - x^{2} \cdot 1$

Multiply $- 1$ by $1$ .

$x^{6} + x^{5} - 3 x^{5} - 3 x^{4} + 3 x^{4} + 3 x^{3} - x^{3} - x^{2}$

Simplify by adding terms.

Combine the opposite terms in $x^{6} + x^{5} - 3 x^{5} - 3 x^{4} + 3 x^{4} + 3 x^{3} - x^{3} - x^{2}$ .

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

$x^{6} + x^{5} - 3 x^{5} + 0 + 3 x^{3} - x^{3} - x^{2}$

Add $x^{6} + x^{5} - 3 x^{5}$ and $0$ .

$x^{6} + x^{5} - 3 x^{5} + 3 x^{3} - x^{3} - x^{2}$

Subtract $3 x^{5}$ from $x^{5}$ .

$x^{6} - 2 x^{5} + 3 x^{3} - x^{3} - x^{2}$

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

$x^{6} - 2 x^{5} + 2 x^{3} - x^{2}$

The largest exponent is the degree of the polynomial.

$6$

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.

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

Simplify terms.

Simplify each term.

Multiply $- 1$ by $3$ .

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

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

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

Multiply $3$ by $1$ .

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

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

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

Apply the distributive property.

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

Simplify.

Multiply $x^{2}$ by $x^{3}$ by adding the exponents.

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

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

Add $2$ and $3$ .

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

Rewrite using the commutative property of multiplication.

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

Rewrite using the commutative property of multiplication.

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

Move $- 1$ to the left of $x^{2}$ .

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

Simplify each term.

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

Move $x^{2}$ .

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

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

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

Add $2$ and $2$ .

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

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

Move $x$ .

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

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

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

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

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

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

Add $1$ and $2$ .

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

Rewrite $- 1 x^{2}$ as $- x^{2}$ .

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

Expand $(x^{5} - 3 x^{4} + 3 x^{3} - x^{2}) (x + 1)$ by multiplying each term in the first expression by each term in the second expression.

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

Simplify terms.

Simplify each term.

Multiply $x^{5}$ by $x$ by adding the exponents.

Multiply $x^{5}$ by $x$ .

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

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

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

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

Add $5$ and $1$ .

$x^{6} + x^{5} \cdot 1 - 3 x^{4} x - 3 x^{4} \cdot 1 + 3 x^{3} x + 3 x^{3} \cdot 1 - x^{2} x - x^{2} \cdot 1$

Multiply $x^{5}$ by $1$ .

$x^{6} + x^{5} - 3 x^{4} x - 3 x^{4} \cdot 1 + 3 x^{3} x + 3 x^{3} \cdot 1 - x^{2} x - x^{2} \cdot 1$

Multiply $x^{4}$ by $x$ by adding the exponents.

Move $x$ .

$x^{6} + x^{5} - 3 (x \cdot x^{4}) - 3 x^{4} \cdot 1 + 3 x^{3} x + 3 x^{3} \cdot 1 - x^{2} x - x^{2} \cdot 1$

Multiply $x$ by $x^{4}$ .

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

$x^{6} + x^{5} - 3 (x^{1} x^{4}) - 3 x^{4} \cdot 1 + 3 x^{3} x + 3 x^{3} \cdot 1 - x^{2} x - x^{2} \cdot 1$

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

$x^{6} + x^{5} - 3 x^{1 + 4} - 3 x^{4} \cdot 1 + 3 x^{3} x + 3 x^{3} \cdot 1 - x^{2} x - x^{2} \cdot 1$

Add $1$ and $4$ .

$x^{6} + x^{5} - 3 x^{5} - 3 x^{4} \cdot 1 + 3 x^{3} x + 3 x^{3} \cdot 1 - x^{2} x - x^{2} \cdot 1$

Multiply $- 3$ by $1$ .

$x^{6} + x^{5} - 3 x^{5} - 3 x^{4} + 3 x^{3} x + 3 x^{3} \cdot 1 - x^{2} x - x^{2} \cdot 1$

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

Move $x$ .

$x^{6} + x^{5} - 3 x^{5} - 3 x^{4} + 3 (x \cdot x^{3}) + 3 x^{3} \cdot 1 - x^{2} x - x^{2} \cdot 1$

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

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

$x^{6} + x^{5} - 3 x^{5} - 3 x^{4} + 3 (x^{1} x^{3}) + 3 x^{3} \cdot 1 - x^{2} x - x^{2} \cdot 1$

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

$x^{6} + x^{5} - 3 x^{5} - 3 x^{4} + 3 x^{1 + 3} + 3 x^{3} \cdot 1 - x^{2} x - x^{2} \cdot 1$

Add $1$ and $3$ .

$x^{6} + x^{5} - 3 x^{5} - 3 x^{4} + 3 x^{4} + 3 x^{3} \cdot 1 - x^{2} x - x^{2} \cdot 1$

Multiply $3$ by $1$ .

$x^{6} + x^{5} - 3 x^{5} - 3 x^{4} + 3 x^{4} + 3 x^{3} - x^{2} x - x^{2} \cdot 1$

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

Move $x$ .

$x^{6} + x^{5} - 3 x^{5} - 3 x^{4} + 3 x^{4} + 3 x^{3} - (x \cdot x^{2}) - x^{2} \cdot 1$

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

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

$x^{6} + x^{5} - 3 x^{5} - 3 x^{4} + 3 x^{4} + 3 x^{3} - (x^{1} x^{2}) - x^{2} \cdot 1$

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

$x^{6} + x^{5} - 3 x^{5} - 3 x^{4} + 3 x^{4} + 3 x^{3} - x^{1 + 2} - x^{2} \cdot 1$

Add $1$ and $2$ .

$x^{6} + x^{5} - 3 x^{5} - 3 x^{4} + 3 x^{4} + 3 x^{3} - x^{3} - x^{2} \cdot 1$

Multiply $- 1$ by $1$ .

$x^{6} + x^{5} - 3 x^{5} - 3 x^{4} + 3 x^{4} + 3 x^{3} - x^{3} - x^{2}$

Simplify by adding terms.

Combine the opposite terms in $x^{6} + x^{5} - 3 x^{5} - 3 x^{4} + 3 x^{4} + 3 x^{3} - x^{3} - x^{2}$ .

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

$x^{6} + x^{5} - 3 x^{5} + 0 + 3 x^{3} - x^{3} - x^{2}$

Add $x^{6} + x^{5} - 3 x^{5}$ and $0$ .

$x^{6} + x^{5} - 3 x^{5} + 3 x^{3} - x^{3} - x^{2}$

Subtract $3 x^{5}$ from $x^{5}$ .

$x^{6} - 2 x^{5} + 3 x^{3} - x^{3} - x^{2}$

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

$x^{6} - 2 x^{5} + 2 x^{3} - x^{2}$

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

$x^{6}$

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)^3(x+1)

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